Mauno Rönkkö
email: mauno.ronkko@uef.fi

Ville Kotovirta
email: ville.kotovirta@vtt.fi

Mikko Kolehmainen
email: mikko.kolehmainen@uef.fi

Classifying Environmental Monitoring Systems

Keywords: Environmental information systems, Environmental monitoring, Systems Architecture, Systems Analysis, Systems Classification

Introduction

There are currently many factors driving the design and development of both national and international environmental information systems. One of the most notable of these factors is the green house effect. Reducing energy consumption, cutting greenhouse gas emissions and eliminating energy wastage are of global interest. Another significant factor has to do with the vulnerability of modern society in case of natural disasters. This has also been highlighted by recent extreme weather phenomena exhibited around the globe.

Not surprisingly, as pointed out by Messer et al. [START_REF] Messer | Environmental monitoring by wireless communication networks[END_REF] "High-resolution, continuous, accurate monitoring of the environment is of great importance for many applications from weather forecasting to pollution regulation." Environmental monitoring has, thus, become an attractive topic for state-of-the-art research. For instance, monitoring of the Fukushima incident is still relevant for impact assessment as discussed by Cyranoski et al. [START_REF] Cyranoski | Fukushima impact is still hazy[END_REF]. Similarly, continuous monitoring of factories and power plants is critical as they may cause long term effects [START_REF] Gilbert | Drug waste harms fish[END_REF].

As the number of new environmental monitoring systems grows, it becomes harder to understand the similarities and differences between the systems. In particular, it becomes harder to understand what the differences mean in terms of future applicability and interoperability. Without this critical information, there is a temptation to develop a new system rather than to adapt and use an existing one; thus, making the overall problem worse.

To make sense of the ever growing number of environmental monitoring systems, a method for the classification is proposed in this article. The method identifies three strategic viewpoints: application domain, functionality, and architecture. The proposed method is used in this article to classify a selected set of environmental monitoring systems. Even for a limited case, the results identify some relevant and interesting clusters. Clustering reveals clearly, which properties are shared by the systems and which properties differentiate a specific system from the rest of the systems within a cluster.

This article is organized as follows. In Section 2 we present the proposed classification method. In Section 3 we describe the environmental monitoring systems considered for classification, and present the classification results. Finally in Section 4 follows the conclusion.

Proposed classification method

There are three strategic viewpoints that are necessary for classification: application domain, functionality, and architecture. The application domain viewpoint describes not only the purpose and need for the use of the system, but also the domain of interest. The functionality viewpoint describes what the system does to satisfy the needs for its use. The architecture viewpoint describes how the system is implemented to provide its functionality. These viewpoints are intentionally orthogonal, to support the analysis process. However, even for these three viewpoints, there are many properties of interest. Therefore, the analysis is limited here to specific properties per viewpoint, as described next.

Application domain viewpoint emphasizes the purpose and the need for the system, including the domain of interest. From this viewpoint there are four kinds of properties of interest:

1. Phenomenon: What is the monitored environmental phenomenon? 2. Interest: Why the end-user is interested in the phenomenon? 3. Task: What is the system used for? 4. Benefit: How do the end-users benefit from using the system? Functionality viewpoint emphasizes the activities of the system that satisfy the end-user's need. From this viewpoint, there are six properties of interest:

1. Operation: What operation is performed by the system? 2. Interaction: How is the operation initiated and carried out? 3. Performance: What kind of performance is expected from the operation? 4. Reliability: How dependable the operation is expected to be? 5. Methods: What methods are used to perform the operation? 6. Maintenance: What kind of maintenance is expected?

Architecture viewpoint emphasizes the details of the system that realize the activities. From this viewpoint, there are four properties of interest:

1. Implementation: How is the system implemented? 2. Data: What kind of data is used by the system? 3. Capacity: What kind of performance can the system provide? 4. Connectivity: How is the system connected to other systems?

The actual analysis is performed by cross-tabulation over the properties for each viewpoint, where each cell indicates if the considered system supports the combined properties. This results in three tables as shown in Figures 1,3, and 5. Consequently, the tables provide a qualitative analysis, a comparison over the considered systems. The tables show three different kinds of clusters: properties that are unique to some system, properties that are shared among a few systems, and properties that are common to most of the systems. Properties that are unique to a system indicate the competitive edge of the system. Properties that are shared among a few systems indicate a higher level feature that is somehow central to those systems. Lastly, properties that are common to most of the systems indicate compulsory requirements. Thus, meaningful clusters are found by finding clusters of properties that are shared by only a few systems.

Results

In this section, environmental monitoring systems that were classified are introduced first shortly. A set of systems were chosen to represent different kinds of environmental monitoring systems. The classification results and discussion are presented at the end of this section.

Environmental monitoring systems to be classified

Measuring and forecasting. For this class of systems, NOAA Integrated Ocean Observing Systems (IOOS) was considered. As presented by Harlan et al. [START_REF] Harlan | National high frequency radar network: Update[END_REF], the NOAA IOOS uses a high frequency radar network for measuring ocean surface currents in United States. It consists of more than 100 high frequency radars. As pointed out by Harlan et al., the radar network constitutes the largest set of ocean current observations in existence.

Integrated service. For this class of systems, a weather service application integrating Weather Underground and INTAMAP [START_REF] Williams | Supporting interoperable interpolation: the intamap approach[END_REF] was considered, later called INTAMAP WU. As presented by Williams et al. [START_REF] Williams | Automatic processing, quality assurance and serving of real-time weather data[END_REF], the application provides quality control for weather monitoring over a collective network of privately owned weather stations. It provides an OGC standard Observations & Measurements (O&M) interface and uses UncertML [START_REF] Williams | Uncertainty markup language (UncertML)[END_REF] for describing uncertainties for the Weather Underground data.

Household monitoring. For this class of systems, a system for measuring and monitoring indoor air quality and energy efficiency, AsTEKa, by Skön et al. [START_REF] Skön | Energy consumption and air quality monitoring system[END_REF] was considered. It consists of a sensor network for measurements and data capture, a database for data storage, and an application server for monitoring services and support for external user interface built on top of a web browser. The sensor network gathers data for instance about humidity, temperature, and carbon dioxide, as well as data about water and electricity consumption.

Environmental Safety and Security. For this class of systems, a distributed service network for security applications, TiTiMaKe [START_REF] Rönkkö | Titimake final report[END_REF], was considered. According to Rönkkö et al., TiTiMaKe integrates heterogeneous services, including a dedicated sensor network, vehicle detection and classification service, reachability computation and analysis service, chemical dispersion computation and modeling service, and a spatio-temporal population modeling service. TiTi-MaKe implements safety and security applications for traffic monitoring, and estimating affected people and risk groups in case of a chemical accident.

Navigation and planning. For this class of systems, Icebreaker Plot (IB-Plott) [START_REF] Berlund | A system for icebreaker navigation and assistance planning using spaceborne sar information in the baltic sea[END_REF] was considered. IBPlott is a system for icebreaker navigation and assistance planning. It delivers near real-time ice, weather, and hydrological data to users making routing decision in dynamic environmental conditions. The system takes satellite images, weather observations and forecasts, water level observations and forecasts, ice charts and ice model forecasts, as well as ship traffic information as an input.

Participatory sensing. For this class of systems, EnviObserver [START_REF] Kotovirta | Participatory sensing in environmental monitoring experiences[END_REF] was considered. EnviObserver uses people as sensors in collecting observations about the environment. Users use a specialized mobile application to report environmental phenomena and the observations. Time of the observations and the location of the users are stored in the server for later analyses. The observations collected can be viewed graphically near real-time, and alerts can be generated.

Classification results for application domain

The cross-tabulation over application domain properties are presented in Figure 2. There are four distinct clusters of systems. The clusters are health and comfort, sea and water, improved security, and improved efficiency.

Health and comfort. INTAMAP WU, AsTEKa, and EnviObserver forms the first cluster. The shared properties have to do with measuring, learning, and analysis related to comfort, health, and well-being for citizens and households. There is also a strong emphasis on delivering personalized information to citizens.

Sea and water. NOOA IOOS HFR, IBPlott, and EnviObserver form the second cluster. This cluster targets the services for authorities for monitoring and decision making. These systems share strictly the interest of water, sea or lake, as the application domain.

Improved security. OOA IOOS HFR, TiTiMaKe, IBPlott, and EnviObserver form the third cluster. This cluster subsumes the cluster for sea and water; however, these systems share strongly the common interest for improved secu- rity by targeting the services for authorities. There is also a shared interest for understanding the underlying phenomena.

Improved efficiency. OOA IOOS HFR, AsTEKa, and IBPlott form the last cluster. These systems show a strong emphasis on saving costs. Interestingly the tasks involve not only monitoring and decision making, but also learning and analysis.

Classification results for functionality

The cross-tabulation over functionality properties are presented in Figure 4. In the figure, properties unique to the clustered systems are indicated with yellow, and shared properties are indicated with red. There are three distinct clusters of systems. The clusters are local-monitoring, decision making, and robust together with reliable.

Local monitoring. AsTEKa and TiTiMaKe form the first cluster. Both systems perform monitoring of present data with model-based methods. These systems do, however, differ in application domain, which is reflected in the operational properties.

Decision making. NOOA IOOS HFR, INTAMAP WU and TiTiMaKe form the second cluster. In this cluster, shared operations are: producing new data, forecasting, extracting knowledge, and delayed response. The systems differ on data flow and locality of measurements. Also, NOOA IOOS HFR and TiTiMaKe have maintenance that INTAMAP WU does not really have.

Robust and reliable. NOOA IOOS HFR, IBPlott, and EnviObser form the third cluster. The cluster is not cohesive, as only IBPlott and EnviObserver have a subscription service for real-time phenomena monitoring. Moreover, NOOA IOOS HFR and IBPlott have uncertainty management, but only NOOA IOOS HFR and EnviObserver use statistical methods.

Classification results for architecture

The cross-tabulation over architecture properties are presented in Figure 6. In the figure, properties unique to the clustered systems are indicated with yellow, and shared properties are indicated with red. There are four distinct clusters of systems. The clusters are open systems, closed systems, large data flow, and externally dependent.

Open systems. INTAMAP WU and EnviObserver form the first cluster. These systems have loosely coupled, international components and sensors with open interfaces. These two systems differ only by use of programming languages and use of standards.

Closed systems. NOOA IOOS HFR, AsTEKa, and TiTiMaKe form the second cluster. For these systems, use of standards and locality (or nationality) is strongly expressed. In addition, these systems merely provide external interfaces for specific purposes.

Large data flow. OOA IOOS HFR, and IBPlott form the third cluster. Interestingly, these systems share partially the same interest domain, ocean. However, as shown in the cluster, these systems have almost nothing else in common than the use of large data flows.

Externally dependent. INTAMAP WU, TiTiMaKe, IBPlott, and EnviObserver form the last cluster. These systems have loosely coupled components and they also use multiple technologies and programming languages. The cluster is actually divided into to sub-clusters, where IBPlott and EnviObserver focus more on the real-time aspects, and the other two systems focus more on the forecasting aspects.

Discussion

The cross-tabulation revealed interesting clusters for the selected systems. For application domain, a cluster for health and comfort emerged, emphasizing delivery of personalized information. Interestingly, these systems operated at very different domains, which is reflected as unique properties in the cross-tabulation. In the cluster for improved security, the systems shared a strong emphasis for targeting the services for authorities.

The cross-tabulation for functionality revealed a cluster for decision making, where the systems shared the property of delayed response. The system in this cluster differed mostly on having maintenance or not.

The cross-tabulation for architecture revealed a cluster for open systems consisting, emphasizing particularly the use of open interfaces. The architectural similarity in this class could be explained by the fact that the systems are targeted for use by citizens. In the cluster for externally dependent systems, the cross-tabulation clearly reveals how the systems are built for different purposes. In this cluster, the combination of systems sharing some specific property varies substantially from property to property.

Conclusion

The classification method proposed here, produces relevant information about the considered systems, where clusters highlight interests shared by the systems and, at the same time, indicate how such systems differ from each other. Thus, The method is an excellent tool, when choosing a system from among several candidates. It is also a good tool for mapping competition, and strengthening an existing system, for instance, by emphasizing differentiating features.

One interesting topic for future research is to apply the classification method to understand qualitative properties of environmental monitoring systems, such as interoperability, reliability, robustness, and resilience. For such investigation, however, it would be necessary to involve the system developers, as required details tend not to be publicly available. For instance, interoperability seems to be a term easily used for any systems offering some external interfaces. However, as the clustering result in this article with respect to functionality viewpoint indicates, external dependability is not an unambiguous measure.

Fig. 1 .

 1 Fig. 1. Cross-tabulation of the properties for application domain. Labeling: A=NOOA IOOS, B=INTAMAP WU, C=AsTEKa, D=TiTiMaKe, E=IBPlott, F=EnviObserver.

Fig. 2 .

 2 Fig.2. Clusters found for the application domain viewpoint, Figure1; the shared properties are indicated with black fill and the disjoint properties with gray fill.

Fig. 3 .

 3 Fig. 3. Cross-tabulation of the properties for functionality. Labeling: A=NOOA IOOS, B=INTAMAP WU, C=AsTEKa, D=TiTiMaKe, E=IBPlott, F=EnviObserver.

Fig. 4 .

 4 Fig. 4. Clusters found for the functionality viewpoint, Figure 3; the shared properties are indicated with black fill and the disjoint properties with gray fill.

Fig. 5 .

 5 Fig. 5. Cross-tabulation of the properties for architecture. Labeling: A=NOOA IOOS, B=INTAMAP WU, C=AsTEKa, D=TiTiMaKe, E=IBPlott, F=EnviObserver.

Fig. 6 .

 6 Fig.6. Clusters found for the architecture viewpoint, Figure5; the shared properties are indicated with black fill and the disjoint properties with gray fill.

Acknowledgements

We wish to thank the Finnish Funding Agency for Technology and Innovation (TEKES) for funding this research. This work has been carried out in TEKES funded CLEEN SHOK programme, "Measurement, Monitoring and environmental Assessment" (MMEA). We wish also to thank CLEEN Ltd. and all our partners in MMEA for collaboration.


Erik De Rooij
email: erik.derooij@deltares.nl

Klaas-Jan Van Heeringen
email: klaasjan.vanheeringen@deltares.nl

Incorporate Sensor Data and Dynamic Modeling for Real Time Control of Sewer Systems

Keywords: Delft-FEWS, Real Time Control, Webservices, SOAP, SCADA systems, dynamic modeling, Sewer Systems, Sensor Web

We discuss how Delft-FEWS is applied to incorporate real-time sensor data and dynamic modeling to allow Real Time Control (RTC) of three sewer systems in the area of the Hoeksche Waard, the Netherlands. The purpose of this project was to reduce sewerage spilling to the surface water by optimizing the available storage in the sewer systems. In this paper we will present how Delft-FEWS makes use of Simple Object Access Protocol (SOAP) webservices to exchange real-time data with multiple sewer systems and their different Supervisory control and data acquisition (SCADA) systems. We will also explain how we combine this with dynamic models, allowing for central automatic control of multiple sewer systems.

Introduction

Over the last decade Delft-FEWS has been applied in a wide range of operational water management and forecasting systems. In this case study Delft-FEWS is used for the first time to integrate dynamic models with automatic system control. This introduces new challenges concerning; real-time availability of sensor data, fail safe mechanisms in case of network interruptions and fast response times to allow for automatic control. In this paper we discuss how we tackled these challenges, experimenting with different mechanisms for exchanging sensor data between the SCADA systems that are operated at the municipalities and the Waste Water Treatment Plants (WWTP) and Delft-FEWS.

Description Delft-FEWS

Delft-FEWS is a proven, real time software infrastructure for operational water management and forecasting. The system is a sophisticated collection of modules designed for building an operational water management system customized to the specific requirements of an organization. The philosophy of the system is to provide an open shell system for managing the operational management process [START_REF] Werner | Delft-FEWS: An open shell flood forecasting system[END_REF][START_REF] Gijsbers | Delft-FEWS: A proven infrastructure to bring data, sensors and models together 650[END_REF]. This shell incorporates a comprehensive set of general data handling utilities, while providing an open interface to a wide range of forecasting models. The modular and configurable nature of the system allows it to be used both in rudimentary systems and in highly complex systems. Delft-FEWS can be deployed as stand-alone environment, as a fully automated client-server application, or as a web-service component within another forecasting system.

Case Study Hoeksche Waard

For this case study the regional water authority Hollandse Delta and five municipalities (Binnenmaas, Cromstrijen, Korendijk, Oud-Beijerland and Strijen) (see Fig. 1) are exploring ways to increase the efficiency of the waste water treatment process and to improve water quality [START_REF] Nooijen | Implementation of central control for multiple sewer systems[END_REF]. Together with Deltares and Delft University of Technology they started three pilot projects aimed to centralize automatic control of the pumps in the sewer systems of the municipalities and the pumps that transfer sewage water to WWTP. Each of the municipalities and the WWTP operate their own SCADA system. These systems collect the data from the sensors within the sewer system and display this to the operators. From these displays the operators can monitor and control their part of the system. The SCADA systems of the different municipalities and WWTPs are not connected. By introducing Delft-FEWS as a central control component we hope to connect the SCADA systems and achieve the following objectives.

The main objective of this project is to reduce combined sewerage overflow (CSO) volumes from the sewer systems to the surface water. This objective is achieved by optimizing the remaining available storage within the sewer system during storm events. Another objective is the reduction of pumping costs and waste water treatment costs [START_REF] Heeringen | Leveraging existing infrastructure for central automatic control of multiple sewer systems[END_REF]. The Netherlands have an extensive sewer network that collects waste water from households, industries and urban overland flow. To transport the waste water through the system towards the WWTPs an extensive network of sewer pumps is maintained. The capacity of the sewer system is limited and during storm events the system can fill up, resulting in harmful sewage spills to surface waters.

Each municipality is responsible for maintaining and operating the pumps within its boundaries, while the regional water authority is responsible for operating the WWTPs. In general the pumps work independently and do not take into account the levels within the rest of the system. Once the sewage exceeds the local start level the pump will commence operation and will continue until the level drops below the local stop level. Each sewer pump is linked to a SCADA system from which it can be monitored and controlled. The SCADA systems of the municipalities and WWTPs are not linked together.

By linking the SCADA systems of the municipalities and WWTPs together and by introducing a dynamic modeling component it becomes possible to optimize the sewage flow throughout the whole system and at the same time allows for an automated centralized control of the system.

To accomplish this webservices are used to link the SCADA systems together and Delft-FEWS is introduced as dynamic modeling component to be the central information system. Data is collected from each SCADA system and is fed to Delft-FEWS, where a RTC plugin calculates the optimal pump settings. These settings calculated from Delft-FEWS are sent back to the SCADA systems in order to control the sewer pumps.

Automatic Control Requirements

The introduction of the 'automatic control' aspect results in extra requirements to ensure the safety of the sewer system and pumps:

• Fail over mechanism: In the case of network interruptions, or on receiving incorrect pump settings the local system must react to this by switching from automated control back to local control to prevent damage to pumps or cause overflowing to surface waters. • Secured against external interference: The system must be protected against external malevolent parties who try to 'hack' into the system and take control of the pump stations. Firstly this should be prevented by using adequate security measures such as: user authentication, limit access to known IP-addresses and communication over HTTPS. Secondly the fail over mechanism should kick in as soon as harmful control settings are passed to a pumping station. • Real-Time data availability: In order for the dynamic models to produce any meaningful control output it is important that the input data is as close to real-time as possible. Also the return period of the calculated control parameters must be well within the system response time. If either of these requirements is not met the sys-tem will fail over. Experience shows that a response time equal to or less than 30 seconds is required.

Creating a Sensor Web with Webservices

As described earlier in this paper, the SCADA systems of the municipalities and WWTPs reside on individual networks not connected in any way. Each municipality is responsible for maintaining and operating its own SCADA system. None of the municipalities are very eager to open up their system to outsiders and relinquish control. In order to create a Sensor Web from these individual and closed systems it is necessary to minimize the required changes and risks to the local systems. By using webservices the impact to the local systems is kept to a minimum:

• Open a communication port over which all HTTP or HTTPS traffic is sent,

• Install a software component locally on each individual SCADA system that enables the SCADA system to interact with the Sensor Web.

During the course of the project two mechanisms are implemented for transferring data between the SCADA systems and the central controller Delft-FEWS (see Fig. 2):

1. SCADA data is sent to a webservices hosted by Delft-FEWS (FewsPiService).

2. Delft-FEWS interacts with a webservice hosted on the side of the SCADA system. Both of these mechanisms have to be able to interact with the following two SCADA system types:

• Control Maestro: Produced by Elusions SAS (www.elusions.com) and uses OLE for Process Control (OPC) technology. • Flygt (RTiem): Produced by Xylem Nederland (www.xylemwatersolutions.com).

Webservice Hosted by Delft-FEWS

At the start of the project a decision is made to make use of the webservice provided by Delft-FEWS, the FewsPiService1, which exchanges data using the Published In-terface (PI) XML format2. On the SCADA system side this requires the installation of a JAVA process that can exchange data between the SCADA system and the Delft-FEWS webservice. This process is called the Data Transfer Service (DTS) and requires a local configuration file containing information about the data to transfer and at what frequency the data should be transferred. For example one type of data can be scheduled every second while the other is scheduled once a day, at a certain time.

Besides the sending of SCADA data, the DTS also sends log messages produced during data retrieval. Furthermore during network interruptions the DTS stores the data locally in order to re-send it once the network connection is re-established. A schematic representation of this setup can be seen in Fig. 3. Advantages of this approach:

• At the time of implementation, this approach had the advantage of smaller response times, due to the Delft-FEWS system not being able to cope well with model runtimes less than one minute. • Because data is sent at regular intervals it is possible to build a continuous historical archive of the SCADA data. Even during network interruptions the data is cached and is sent later on. This is convenient hence SCADA systems do not always store their historical data.

• By being able to tune the transfer frequencies it is possible to reduce network load and the amount of queries on the SCADA system. • It is easy to manage and send log messages as each process has a dedicated destination.

Disadvantages of this approach:

• Because the DTS process does not run synchronously with the running of the dynamic models, data often arrives slightly too late for the model run or is retrieved slightly before completion of the model run. Resulting in the output control settings lagging in time.

• Because the configuration of the DTS is located on the local SCADA systems, updates always require the assistance of the system operator of the municipalities. This always results in delays. • It is not possible to transfer data using HTTPS, as the Delft-FEWS webservice does not support this.

Webservice Hosted on SCADA System

Later on in the project a decision is made to transfer the control of data exchange to Delft-FEWS. This became possible due to improvements in performance of Delft-FEWS allowing it to run the models as frequent as two to three times a minute. The main reasons for wanting to transfer control are:

• Synchronizing data exchange with model runs.

• Transferring control of configuration to Delft-FEWS For this change a PI webservice is created that is hosted in a Tomcat web container installed on the SCADA system. Now each time Delft-FEWS performs a model run data is first retrieved from the webservice on the SCADA system, then the dynamic model is run and finally the results are sent back to the SCADA system. For each SCADA system an implementation of the new PI webservice is required. Delft-FEWS does not have any dependencies with the SCADA systems as it communicates through the interface hosted by the PI webservice. Now all configurations concerning data transfer and scheduling is controlled by Delft-FEWS. A schematic representation of this setup can be seen in Fig. 4. Advantages of this approach:

• Model run always uses the latest data and returns data as soon as it becomes available. • Configuration resides within Delft-FEWS. It is even possible to update the webservice component through the manager console of Tomcat.

• It is possible to communicate over HTTPS using the security configuration available within Tomcat. • It is still possible to tune the transfer frequencies by scheduling the import process at different rates.

Disadvantages of this approach:

• There is no longer a caching mechanism. So whenever the network connection is interrupted a gap will appear in the historical archive. If the SCADA system allows historical queries of the data the gap can be filled later during a manual import run. • Because it is more flexible to query the SCADA system for historical data (if supported by SCADA), it introduces the risk of OutOfMemory exceptions when querying a too large period. At present no mechanism has been built into the system that splits the results into smaller segments. The only security against this is to report the requested data volume is too large and to return only a partial result set. • It is harder to manage log messages because logged information has to be stored until client retrieves this explicitly in a separate call. If not managed this could lead to memory issues.

Project Results

Prior to this project, the Hoeksche Waard sewer system consisted of numerous individual sensor networks. By linking these networks a greater insight was gained in the actual working of the network as a whole. This revealed that the designed specifications of the sewer system did not correspond to the actual situation. The improved monitoring resulted in a better quantification of the CSO. According to the improved model results there was 48% more CSO than estimated. However because of these new findings it was also possible to implement some simple and cheap improvements.

The goal to reduce pumping costs was not achieved. As a result of the main goal, to reduce emissions, it was not possible to operate the sewer pumps in a more energy efficient way. The only possible solution for reducing pumping costs would be to reduce the inflow of rainwater into the sewer system.

Furthermore flow measurements showed that about two times more water than estimated is entering the sewer system and is being processed by the WWTP's. This leaves a lot of room for improvements and cost reductions.

Conclusions and Future Challenges

This project demonstrates how relatively small adjustments to existing SCADA systems allows them to be converted into centrally controlled operating systems. Currently this project is running as an operational system and plans exist to expand the functionality to other municipalities.

During this project we found that when using a Sensor Web there are inherently going to be network issues or partial system failures. It is therefore important to have a robust backup mechanism in place to assure the integrity of the vital system components.

When transferring SCADA data, package size is usually not a concern, as the amount of data transferred is small. But when it is possible to query larger amounts of data, memory management will become a concern. For the future it would be good to investigate mechanisms that support the transfer of large volumes of data, either by segmenting the results or by streaming them.

Fig. 1 . 2 Centralizing

 12 Fig. 1. Hoeksche Waard area with the five municipalities

Fig. 2 .

 2 Fig. 2. Overview transfer mechanisms

Fig. 3 .

 3 Fig. 3. Schematic representation of webservice hosted by Delft-FEWS

Fig. 4 .

 4 Fig. 4. Schematic representation of webservice hosted on SCADA system

http://publicwiki.deltares.nl/display/FEWSDOC/Fews+PI+service

http://publicwiki.deltares.nl/display/FEWSDOC/The+Delft-Fews+Published+interface+%28PI%29

Acknowledgments. This project was supported by grants from Agentschap NL, The Netherlands. This agency provided a subsidy from the "Innovatieprogramma Kaderrichtlijn Water" of the Dutch Ministry of Transport, Public Works and Water Management. This work is part of the project "Integrale Sturing Afvalwaterketen en watersysteem Hoeksche Waard".


Zuzana Bezáková
email: zuzana.bezak@gmail.com

Green Computing Practices as a Part of the Way to the Sustainable

Keywords: sustainable development, sustainable computing, green computing, innovation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Innovation and Sustainable Development

The concept of innovation evokes in many groundbreaking technological inventions that are the result of long-term activities and research work of scientists. However, innovation can occur in any business, and there is an effort to success in the market, be faster than its competitors, and also to offer products that satisfy customers. Innovation need not be oriented only to products but also to services, or processes, or marketing practices [START_REF] Zaušková | Inovačná schopnosť a inovačná výkonnosť podnikateľských subjektov[END_REF]. Creation and management of innovation processes are included with all its principles in the full range of corporate governance [START_REF] Zaušková | Tvorivosť v inovačnom procese[END_REF]. Much attention has recently been devoted to innovation as a means of achieving a radical improvement of the system of corporate environmental practices and performance. Eco-innovation is perceived as key concept that provides direction and vision for tracking overall societal changes necessary to achieve sustainable development [START_REF]Sustainable Manufacturing and Eco-Innovation[END_REF].

We can harness the power of IT to solve environmental problems but also try to make our IT systems and their use greener. Green IT, also known as green computing is the study and practice of designing, manufacturing and use of computers, servers, monitors, printers, storage devices, and networking and communications systems effectively and efficiently, with no or minimal impact on the environment. Green IT is used as a support for other environmental activities and help to create green awareness. Thus, green IT includes hardware, software, tools, strategies and practices that improve and promote environmental sustainability.

Green IT benefits the environment by improving energy efficiency, reducing greenhouse gas emissions, using less harmful materials, the re-use and recycling is also important. Thus, green IT includes the dimension of environmental sustainability, economics energy efficiency and total cost of ownership, which includes the cost of disposal and recycling. Increased awareness of the harmful effects of greenhouse gas emissions, new stringent environmental legislation, concerns about procedures for handling electronic waste and concern about corporate image are the driving force for businesses and individuals to become green [START_REF] Murugesan | Harnessing Green It[END_REF].

Currently available sources talking about sustainability present three broad environmental objectives leading to sustainability:

1. Eco-efficiency -the supply of goods and services at competitive prices to satisfy human needs and bring quality of life, while progressively reducing ecological impacts and resource intensity throughout the life cycle in accordance with the earth's carrying capacity 2. Eco-equity -means equality between people and generations, and equal rights of all people to natural resources 3. Eco-effectiveness -considered the best solution to environmental problems, but requires a shift in thinking and transforming business models. People need to stop environmental degradation and depletion of non-renewable resources [START_REF]Information systems and environmentally sustainable development: Energy informatics and new directions for the IS community[END_REF].

The most significant impacts on the company's ability to behave sustainably include:

 social impact -this is a measurement of the impact of the enterprise in terms of the social contract and the impact of interest groups  environmental impact -the effect of business activities on the geophysical environment  organizational culture -involves the relationship between the company and internal stakeholders, particularly employees, and all aspects of the relationship  finance -are evaluated in terms of adequacy of compensation for assuming the risk.

It is necessary to find a balance between the influences that will ensure not only sustainability, but also sustainable development. The sustainable development is the new view for business, it is a change of the established practices. Addressing sustainability in the business area brings number of sub-issues.

The basic features are:

 sustainable production  sustainable consumption  sustainable marketing  sustainable profit [START_REF] Šmida | Predpoklady budovania spoločensky zodpovedného podnikania ako súčasti udržateľnej spoločnosti v zmysle konceptu udržateľného rozvoja[END_REF].

Green Computing

Green computing can be included in the field of sustainable consumption, in terms of access of business entity to sustainable development. It can be defined as the use of computer resources efficiently in practice. It is important that the initiative of green computing was systemic in nature and was designed to address increasingly sophisticated problems. Green computing refers to the requirements of environmental protection and energy saving together with the reduction of operational costs in today's increasingly competitive world. But before it comes to implementation, it is also important to deal with what kind of energy and performance gains can be achieved through it. Therefore, analysis of what we have today and what we have to do is essential and important for achieving the benefits of green computing [START_REF]Green computing: Practice of Efficient and Eco-Friendly Computing Resources[END_REF].

Governments of various countries are increasingly pushing for the adoption of green computing and sustainable IT investment and sustainable practices. Due to the immediate impact on business value, it is likely that green computing will continue to focus some time to reduce costs while improving the efficiency of data centers and desktops [START_REF]Sustainable IT services: Assessing the impact of green computing practices[END_REF].

IT can be a key factor in greening even other sectors and also the initiator of positive initiatives towards environmental sustainability. Positive change in IT should be reflected in the change of environment. Several studies show that IT contributes only about 3% of greenhouse gas emissions to the total quantity, this means that the vast majority of emissions come from non-IT resources.

Green IT can help create a better environment through these tools, technologies and systems:

 Software tools for analysis, modeling and simulation of environmental impacts and risk management environment  Platform for eco-management, emissions trading and ethical investment  Tools for auditing and reporting energy consumption and savings and for monitoring greenhouse gas emissions  Environmental knowledge management systems, designed for the collection and transmission of environmental knowledge, support systems, environmental ontology  Environmental engineering systems, including geographic information systems  Urban planning tools and systems environment  Technologies and standards for interoperable environment monitoring networks, intelligent sensor networks  Integration and optimization of existing environmental monitoring networks, easy to plug-in new sensors, Sensors cooperation and networking  Tools and systems to optimize business processes in organizations.

To reduce their carbon footprint, organizations can dematerialize some of their products and activities, using IT. Dematerialization refers to the transformation of the physical goods Information goods represented in digital form, so-called "Turning atoms to bits" [START_REF] Murugesan | Harnessing Green It[END_REF].

3

Sustainable Approach to Hardware and Software

Green PCs, Laptops and Servers

Every company can significantly reduce electricity consumption by making small changes in the way the computer is used. Energy cost savings of one computer does not seem to be an important item, but combined savings of hundreds of computers in the enterprise is already considerable and is not negligible. PC power consumption can be reduced by adopting several measures:  Enabling power management features  Turn off the system when not in use  Use a screen saver  The use of thin client computers. These measures will not become a reality without the sincere willingness of users and their active participation. Wise companies tend to adopt innovative environmental strategies, create value and build competitive advantage through innovation in the field of computing [START_REF] Murugesan | Harnessing Green It[END_REF].

Green Data Centers

Green data centers (DC) are those in which the computer system, air conditioning, electrical and mechanical systems, which cover data centers are designed and operated for maximum energy efficiency, low carbon footprint and minimal environmental impacts. Operators who want to optimize the energy efficiency of datacenters, have many options to choose from, but usually cannot know in advance exactly identify potential savings before investing. Also, it is very difficult to determine in advance the actual risk of the new approach in a real environment. It is therefore important that the operator can evaluate DC savings, but also the risks of several options before making a choice. For this purpose there are various simulation systems, such as The EoDD (Energy Optimized Data Center Designer) [START_REF]Capabilities and Limitations of Direct Free Cooling in Data Centers[END_REF].

Performance metrics for green computing data centers can be divided into two categories: basic metrics and advanced metrics.

Basic metrics provide a basic picture of the data center. These include:

 Metrics of greenhouse gas  Metrics of humidity  Thermal metrics  Metrics of Power.

Advanced metrics provide a deeper look at the data centers, and include:

 Multiple indicators  Total cost of ownership.[10]

Green Cloud Computing

Cloud computing delivers infrastructure, platform, and software (applications) as services that are available to consumers in the pay-as-you-go model. In the industry, these services are referred to as Infrastructure as a Service (IaaS), Platform as a Service (PaaS) and software as a service (SaaS). Many providers of computing services, including Google, Microsoft, Yahoo, and IBM quickly deploy data centers in different locations around the world, to be able to deliver cloud computing services [START_REF]Energy-aware resource allocation heuristics for efficient management of data centers for Cloud computing[END_REF]. In the cloud computing world, IT capabilities are delivered on the fly and on-demand through the Internet when the need arises, instead of drawing from desktop computers. Many design and architectural patterns are emerging around cloud computing that make it difficult to fit everything into a perfect definition [START_REF]Green Computing: A Dual Technology for Cloud Computing and HPC[END_REF].

Green Data Storage

Data and information storage requirements are growing constantly. Storage systems in data centers consume large amounts of energy for running and cooling. For example, in a data center, storage systems consume 24% to 40% of total energy consumption of IT and are the largest centers of power. In addition to the requirements for energyefficient servers, there is currently the focus on the greening of data storage. Energy management of data storage can be divided into two basic areas: hard-disk energy management and system-level energy management among the various main storage devices, which include hard disk drives, magnetic tape drives and solid state drives, consume more power than tape and SSD. The main techniques for dealing with the energy of individual disks are transitioning state, caching and dynamic rotations per minute (DRPM). System-level energy management is about system-wide management across different or multiple components or devices. Energy management techniques for redundant array of independent disks (RAID) systems, tiered storages, virtualized storage and cloud storage are in this category. Green data storage is a priority for many European storage professionals, and some of them have begun to implement measures to cut energy usage. Green storage considerations are important when buying storage hardware. Many users would pay extra money for more energy-efficient hardware [START_REF]Green data storage technology survey: Green storage a priority for European storage professionals[END_REF].

Green Software

Green software is the result of innovation process towards sustainability. Innovation is the successful introduction of new product, service, or process. Software innovation can be seen as a process leading to:

 development of a novel aspect, feature or application of an existing software product or process, or  introduction of a new software product, service or process or an improvement in the previous generation of the software product or process, and  entry to an existing market or the creation of a new market [START_REF] Lippoldt | Innovation in the Software Sector[END_REF].

Innovated software is an important element in determining the total energy consumption and computational efficiency. Software is a key element in improving environmental sustainability. Green software is software that is environmentally friendly, and especially helps in improving the environment. Green software can be divided into four categories:  software that is environmentally friendly, uses less energy to run  embedded software that helps other things being green  sustainability reporting software, or carbon management software (CMS)  software for understanding climate change, assess its impact [START_REF] Murugesan | Harnessing Green It[END_REF].

Green Networks and Communication

Networking and communication play more important role than ever before, and facilitate the transfer and sharing of data. It allows us to communicate and share information, trade, learn and socialize online, so that we can more effectively use our working time, and also can do many other things. Green Network shows ways to minimize the impact of networks on the environment through energy-efficient network technologies, protocols and products, and minimize the use of resources. Procedures of green networks include the following:

 the use of newer, more energy-efficient techniques, technologies and products  update of older devices and replacing them with newer, greener network devices  the use of intelligent systems, user management and energy conservation through networks to improve energy efficiency  the use of remote management and video conferencing instead of traveling [START_REF] Murugesan | Harnessing Green It[END_REF].

Sustainable Computing in the Slovak Business Environment

We have made the research dedicated to the point of view of Slovak business entities to sustainable computing. The aim of our research was to find out what are the attitudes of Slovak companies to sustainability but mainly to sustainable computing /sustainable use of PC/. We also examined whether there are specific activities in this area, to which the companies are dedicated. The results should give us an idea of the current state, so we can propose solutions to improve the routing of these companies towards sustainable computing and thereby protect the environment. Also, we wonder whether sustainable computing is an important factor from the point of view of the competitiveness in Slovak business environment and what is the position of the customer in this process.

The survey included 45 Slovak IT companies, advertising agencies and web hosting companies that deal with domain registration, web hosting, online marketing, network management, network solutions, etc.

We focused on small /80%/ and medium enterprises /20%/. The reason was, that [START_REF] Čábyová | Benchmarkingové porovnávanie v oblasti marketingovej komunikácie[END_REF] in Slovakia recently grows importance of small companies. Their advantage is that they mostly focus on one segment, which can adapt well their marketing mix. 79% of respondents consider their business as sustainable, but this behavior is rather linked with activities such as recycling paper, waste separation, use of electronic invoices and so. We also found that most companies /80%/, which indicate their behavior as sustainable, use the devices -PC, network solutions that are friendly to the environment.

Survey results indicate that these efforts are related to the saving of electricity, which is beneficial to society, but of course it has a positive impact on the environment.

Eco attribute of the device for which they decide in purchase are in favor of ecologic devices only in 27%. Another respondents do not consider this fact as important /33%/, while others argue that the price is mainly important when buying a particular device /33%/.

As mentioned in the article, the energy consumption can be reduced through simple procedures. Most companies /93%/ use at least one procedure to reduce electricity consumption, including the most frequently used such as screen saver, standby PC when it is not in use, or even a complete system shutdown. Just in one case they intentionally use the thin client computers or laptops that have lower power consumption compared to PC.

The status of the customer in the process of sustainable computing is very important. The term "Green Company" is not seen as a big advantage /73%/ in the Slovak business environment.

Customers are not actively interested in environmental aspects of solutions /87%/, Slovak companies meet this interest very rarely /13%/.

Websites are important for the presentation, and also in attracting new customers. The trend is that larger companies are dedicated to their websites and presentation of sustainable behavior, social responsibility, thus helping to improve the environment. Even small and medium-sized companies in our survey indicate in some cases their green activities on websites /27%/, but for many of them it is not at all important to give this information to the customer /47%/. Some companies plan to publish such information on the website /13%/. Of course, companies that do not perform any environmental activities, cannot advertise them /13.3%/.

Conclusions

Sustainable IT services are essential for the success in business. They not only focus on data center efficiency and on minimizing carbon emissions. They are aimed direct-ly at the long-term importance of IT for the organization, its customers and for the whole society. Therefore sustainable computing refers to everything what the organization needs to do in order to ensure that IT services will provide excellent value to achieve a strong market position and ability to secure its survival. It is important to align IT with the business strategy to achieve market-leading business value, customer value and societal value. This will ensure the viability of IT within the organization. We define sustainable social services in total value terms as the total amount that is available to the company. Systematic integration and alignment of IT services is important in terms of creating greater social value [START_REF]Sustainable IT services: Assessing the impact of green computing practices[END_REF]. Companies should know wishes and needs of customers. It is important to have enough information about them and then use them in their business [START_REF] Matúš | Marketing -základy a nástroje[END_REF].

The results of the survey show that Slovak companies are trying to implement green approach in the use of hardware and sustainable computing /sustainable use of computers/. Companies themselves do not consider this fact as a competitive advantage in the Slovak business environment. They try to use devices that are environmentally friendly, or trying to use simple procedures to reduce electricity. Also, quite a lot make extensive use of other technical equipment, but also networking solutions that are environmentally friendly.

Based on the survey results, we can conclude that the customer is not an important factor in sustainable computing in Slovakia. The customer is not very interested in greener solutions which are probably the reason why there is even a small degree of companies' promotion of their green activities on the site.

An interesting solution and recommendation is:

 Draw customers into the green activities of the company, gradually acquaint with the importance of sustainable behavior.  The use of organic labeling and explaining the importance of the environmental behavior importance may lead to an overall improvement of the environment.  More frequent use of advertising "slogans" with the sustainable theme can help you to become more and more to receive this concept to the attention of the customer  To offer organic solutions and services and IT solutions, offer sustainable computing initially only as an added value.

These measures can help in the Slovak business environment to build a solid foundation for sustainable use of computers, strengthen the importance of this concept for companies, but especially the need to build sustainable behavior of the customer.

Acknowledgements. This article has been elaborated within the project VEGA 1/0900/12 -"Improving innovation performance and innovation entities through a system of open innovation with support of integrated marketing communication".


Jean-François Grosjean
email: grosjean@iecn.u-nancy.fr

MINIMAL SUBMANIFOLDS WITH A PARALLEL OR A HARMONIC p-FORM

Keywords:

The purpose of this paper is to study the relations between the existence of minimal immersions of a Riemannian manifold M into another and some structural or topological properties of M . The properties on M which we consider are the existence of a parallel or a harmonic p-form.

Introduction

The purpose of this paper is to obtain some non existence results about minimal submanifolds. Let (M m , g) be an m-dimensional Riemannian manifold isometrically immersed by φ in an n-dimensional Riemannian manifold (N n , h) (n > m). The Gauss equation allows us to obtain rigidity results in terms of geometry of (M m , g) and (N n , h). For example, as a first consequence of the Gauss equation, we get the following well known inequality in each point x of (M m , g)

|H(x)| 2 ≥ (1/m)((Scal(x)/m) -(m -1)K 1 (φ(x))) (1)
where |H(x)| 2 and Scal(x) are respectively the square of the mean curvature of φ and the scalar curvature of (M m , g) at x and K 1 (φ(x)) is the largest sectional curvature of (N n , h) at φ(x). In particular, if K 1 has an upper bound and if Scal > m(m -1)K

1 max
(where K

1 max = max N (K 1
)) for at least a point of (M m , g), there is no minimal immersion of (M m , g) into (N n , h). Many other results were obtained, by assuming that (M m , g) is endowed with some particular structures or topological properties (see for instance [START_REF] Carlson | Harmonic mappings of Kaehlerian manifolds to locally symmetric spaces[END_REF], [START_REF] Sampson | Harmonic maps in Kähler geometry[END_REF] and [START_REF] Dajczer | Rigidity of a real Kaehler submanifolds[END_REF]). First recall the results of Sampson ([12]) and Dajczer and Rodriguez ([3]). They proved that there is no minimal immersion of an m-dimensional Kaehlerian manifold (m ≥ 4) into a Riemannian manifold of negative constant sectional curvature. Later, El Soufi ( [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF]) obtained a generalization of this result by assuming a pinching of the sectional curvature of (N n , h) and Hernandez ([9]) obtained the same conclusion under the negativity of the complex sectional curvature of (N n , h). More recently, Petit and El Soufi ([6]) extend this result in the case where (M m , g) is not necessarily Kaehlerian but has a parallel 2form and where the isotropic curvature of (N n , h) is negative (recall that the isotropic curvature of a Riemannian manifold is the restriction of the complex sectional curvature to isotropic tangent planes ( [START_REF] Micallef | Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two planes[END_REF])).

The section 1 of the present paper deals with some preliminaries. In the section 2, we consider the general case where (M m , g) has a parallel p-form and we prove (theorem 3.1) that if (N n , h) satisfies a curvature pinching condition, then there is no minimal immersion from (M m , g) into (N n , h). This is the generalization of the result of El Soufi stated in [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF] for the case where (M m , g) is Kaehlerian. Note that this theorem as well as the other results recalled above are of interest only if the sectional curvature of (N n , h) is negative. However, in the theorem 3.2, we obtain the same conclusion with a new pinching condition for the case where (N n , h) is not necessarily of negative sectional curvature but has a negative smallest sectional curvature. In the theorem 3.3, we study the particular case where (N n , h) is the complex hyperbolic space CH n (c) with constant holomorphic curvature equal to c and we prove that there is no totally real minimal immersion of a Riemannian manifold (M m , g) with a parallel p-form into CH n (c).

The compact manifolds with a parallel p-form are a particular case of manifolds with a harmonic p-form (or a nonzero p-th Betti number b p (M )). In the section 3, we prove (theorem 4.1 and theorem 4.2) that for any compact manifold (M m , g) with b p (M ) = 0 and isometrically immersed in a Riemannian manifold (N n , h), there exists at least a point

x of M so that m √ p p -1 p |B(x)||H(x)| ≥ k(x) - p -1 p ((m -1)K 1 + ρ 1 )(φ(x))
and

m p -1 √ p + m -p -1 √ m -p |B(x)||H(x)| ≥ Scal(x) -(m -2)((m -1)K 1 + ρ 1 )(φ(x))
where |B(x)|, k(x) and ρ 1 (φ(x)) denote respectively the norm of the second fundamental form of φ, the smallest eigenvalue of the Ricci curvature of (M m , g) at x and the largest eigenvalue of the curvature operator of (N n , h) at φ(x). El Soufi proved the first inequality for p = 2 in [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF] and the second for p = 2 but only for m = 4. The first is optimal for the usual standard minimal embeddings of the Clifford torus and of the complex projective space in the sphere. These inequalities will be a consequence of a new lower bound of the curvature term in the Weitzenböck formula for p-forms (see the relation [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF] and the propositions 4.1 and 4.2). As a consequence of the previous inequalities, we deduce

(corollary 4.2) that if (M m , g) is minimally immersed in (N n , h), if ρ 1 is bounded above and if min M (Scal) > (m -2) (m -1) max N (K 1 ) + max N (ρ 1 )
then (M m , g) is a sphere of homology. This result can be viewed as a generalization of a theorem of Leung ([10]) which has shown that if a compact Riemannian manifold (M m , g) is minimally immersed in a unit sphere and if the scalar curvature satisfies Scal > m(m-2) then it is homeomorphic to an m-dimensional sphere.

Preliminaries and notations

Let (M m , g) be an m-dimensional Riemannian manifold and let φ be an isometric immersion of (M m , g) into an n-dimensional Riemannian manifold (N n , h) (n > m). The inner product and the norm induced by g and h on the tensors will be denoted respectively by , and | | 2 . Moreover, we denote respectively by R, ρ, Ric and Scal the curvature tensor, the curvature operator, the Ricci tensor and the scalar curvature of (M m , g) and by R, K, ρ and Scal the curvature tensor, the sectional curvature, the curvature operator and the scalar curvature of (N n , h). We recall that for all vector field X, Y , Z, W ∈ Γ(T N ), ρ is defined by

ρ(X ∧ Y ), Z ∧ W = R(X, Y, Z, W )
Moreover, for all vector field X, Y ∈ Γ(T M ), the tensor R φ will be given by

R φ (X, Y ) = i≤m R(dφ(X), dφ(e i ), dφ(Y ), dφ(e i ))
where (e i ) 1≤i≤m is an orthonormal frame on M . On the other hand, for all x ∈ N , we denote respectively, K 1 (x) and K 0 (x) the largest sectional curvature and the smallest sectional curvature at x and ρ 1 (x) and ρ 0 (x) the largest eigenvalue and the smallest eigenvalue of the curvature operator. Then, it is easy to see that

ρ 0 (x) ≤ K 0 (x) ≤ K 1 (x) ≤ ρ 1 (x) (2)
Now, let B be the second fundamental form of the immersion φ and let H be the mean curvature vector defined by: H = (1/m)trace B. The Gauss equation tells us that for any vector field X, Y , Z, W ∈ Γ(T M ), we have

R(X, Y, Z, W ) = R(X, Y, Z, W ) + B(X, Z), B(Y, W ) -B(X, W ), B(Y, Z) (3)
For the sake of completeness, we need now to recall briefly some definitions and properties about p-forms. Let (e i ) 1≤i≤m be a local orthonormal frame. Throughout this paper, for all q-tensor T , we will write T i 1 ...iq instead of T (e i 1 , ..., e iq ) and then the inner product of two p-forms ω and θ of (M m , g) will be

ω, θ = 1 p! 1≤i 1 ,...,ip≤m ω i 1 ...ip θ i 1 ...ip
The inner product (or contraction) i(X)ω of a p-form ω with a vector field X on M is a p -1-form, defined by

(i(X))ω(X 1 , ..., X p-1 ) = ω(X, X 1 , ..., X p-1 ), ∀ X 1 , ..., X p-1 ∈ Γ(T M )
More generally, if X 1 , ...X q ∈ Γ(T M ), then the inner product of the p-form ω with the q-tensor X 1 ∧ ... ∧ X q is the p -q-form defined by

(i(X 1 ∧ ... ∧ X q )ω) (Y 1 , ..., Y p-q ) = ω(X q , ..., X 1 , Y 1 , ..., Y p-q ), ∀ Y 1 , ..., Y p-q ∈ Γ(T M )
Recall some elementary facts about inner and exterior products. Let ω and θ be respectively a p-form and a q-form and let X be a vector field on M , then

i(X)(ω ∧ θ) = i(X)ω ∧ θ + (-1) p ω ∧ i(X)θ
and if X ⋆ is the dual 1-form of the vector field X with respect to g, then i(X) is in fact the adjoint of left exterior multiplication by X ⋆ , that is

i(X)(ω), θ = ω, X ⋆ ∧ θ
If M is orientable, we also need the following relation between the inner product and the Hodge operator ⋆ on (M m , g) (see for instance [START_REF] Abraham | Manifolds, Tensor Analysis and applications[END_REF])

i(X)(⋆ω) = (-1) p ⋆ (X ⋆ ∧ ω)
On the other hand, if α is a 1-form which is real valued and β is a 1-form which is valued in a vector bundle, we define the 2-tensor α ∨ β by

(α ∨ β)(X, Y ) = α(X)β(Y ) + α(Y )β(X) (4)
We denote now by d, d ⋆ , ∇ and ∇ ⋆ respectively the exterior differential and the codifferential acting on p-forms, the covariant derivative of (M m , g) extended to p-forms and its adjoint with respect to g. The Hodge-de Rham Laplacian ∆ acting on p-forms is given by

∆ω = dd ⋆ ω + d ⋆ dω
To compare this Laplacian to the "rough" Laplacian ∇ ⋆ ∇, one has the Weitzenböck formula, reading as

∆ω = ∇ ⋆ ∇ω + R p (ω), ∀ ω ∈ Λ p (M )
Here R p ∈ End(Λ p (M )) is a bundle endomorphism, given by

R p (ω)(X 1 , ..., X p ) = ij (-1) i [R(e j , X i )ω](e j , X 1 , ..., X i , ..., X p ), ∀X 1 , ..., X p ∈ Γ(T M )
where (e i ) 1≤i≤m is a local orthonormal frame and

R(X, Y )ω = ∇ [X,Y ] ω -[∇ X , ∇ Y ]ω ∀ X, Y ∈ Γ(T M )
An easy consequence of the Weitzenböck formula is the following

1 2 ∆|ω| 2 = ∆ω, ω -|∇ω| 2 -R p (ω), ω (5)
In the sequel, we need to explicit the expression of R p . A straightforward calculation gives us

R p (ω), ω = i,j Ric ij i(e i )ω, i(e j )ω - 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω (6)
and the last term is zero when p = 1.

Geometry of submanifolds having a parallel p-form

The first result of this section is the following theorem

Theorem 3.1 Let (M m , g) be an m-dimensional Riemannian manifold admitting a non- trivial parallel p-form (1 ≤ p ≤ m) and let (N n , h) be an n-dimensional Riemannian manifold (n > m). If for any x ∈ (N n , h), we have (m -1)K 1 (x) < (p -1)ρ 0 (x) (7)
then, there is no minimal immersion from (M m , g) into (N n , h).

Remark 3.1:

1. For p = 2 and for even dimensional manifold (M m , g), the pinching condition [START_REF] Gallot | Opérateur de courbure et Laplacien des formes différentielles d'une variété riemannienne[END_REF] can be refomulate as the negativity of the isotropic curvature ( [START_REF] Soufi | Immersions minimales et immersions pluriharmoniques entre variétés riemanniennes: résultats de non existence et de rigidité[END_REF]). If (M m , g) is Kaehlerian, this condition ( 7) is nothing but that obtained by El Soufi in [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF] (theorem 2.2).

2. From the relation (2), we see that this theorem is of interest only if the sectional curvature of (N n , h) is negative. For the hyperbolic space H n , the condition ( 7) is always satisfied for p < m and then there is no minimal immersion of a manifold having a parallel p-form (1 ≤ p ≤ m -1) into H n . However, the embeddings of H m in H n (m < n) are totally geodesic, and taking the volume form of H m , we see that [START_REF] Gallot | Opérateur de courbure et Laplacien des formes différentielles d'une variété riemannienne[END_REF] is not satisfied for p = m.

In the following theorem, we obtain the same conclusion as in the theorem 3.1 with a new pinching condition where (N n , h) is not necessarily of negative sectional curvature (in fact, there is no condition on K 1 ).

Theorem 3.2 Let (M m , g) be an m-dimensional Riemannian manifold admitting a nontrivial parallel p-form (1 ≤ p ≤ m) and let (N n , h) be an n-dimensional Riemannian manifold (n > m). If for any x ∈ (N n , h), we have

Scal(x) < (n -m)(n + m -1)K 0 (x) + (p(p -1) + (m -p)(m -p -1)) ρ 0 (x) (8)
then, there is no minimal immersion from (M m , g) into (N n , h).

Remark 3.2: We will see in the proof that this theorem is of interest only if the smallest sectional curvature is negative that is K 0 (x) < 0 for all x ∈ N . For instance, let us consider the space N n = H r × S s where n = r + s. Then Scal = -r(r -1) + s(s -1),

ρ 1 = K 1 = 1 and ρ 0 = K 0 = -1. Now, let (M m , g) be a Riemannian manifold of even dimension m and let p = m/2. Then we have Scal-(n-m)(n+m-1)K 0 -(p(p-1)+(m-p)(m-p-1))ρ 0 = 2(s 2 +rs-s-m 2 /4
) and it is easy to see that if r and m are great enough (for instance for a fixed s put m = r great enough) then the condition ( 8) is satisfied and the conclusion of the theorem 3.2 holds for this example.

Proof of theorem 3.1: Let φ be a minimal immersion of (M m , g) into (N n , h) and assume that M has a nontrivial parallel p-form ω. Then R p (ω), ω = 0 and from [START_REF] Soufi | Immersions minimales et immersions pluriharmoniques entre variétés riemanniennes: résultats de non existence et de rigidité[END_REF] we deduce

0 = i,j Ric ij i(e i )ω, i(e j )ω - 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω (9)
and the last term is zero for p = 1. Now from the Gauss formula (3), we obtain i,j

Ric ij i(e i )ω, i(e j )ω = i,j (R φ ) ij i(e i )ω, i(e j )ω + m i,j B ij , H i(e i )ω, i(e j )ω - i,j,k B ik , B jk i(e i )ω, i(e j )ω (10)
and i,j,k,l [START_REF] Micallef | Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two planes[END_REF] by reporting [START_REF] Leung | Minimal submanifolds in a sphere[END_REF] and ( 11) in ( 9), we get

R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω = i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω + i,j,k,l B ik , B jl i(e j ∧ e i )ω, i(e l ∧ e k )ω - i,j,k,l B il , B jk i(e j ∧ e i )ω, i(e l ∧ e k )ω = i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω + 2 i,j,k,l B ik , B jl i(e j ∧ e i )ω, i(e l ∧ e k )ω
0 = i,j (R φ ) ij i(e i )ω, i(e j )ω + m i,j B ij , H i(e i )ω, i(e j )ω - i,j,k B ik , B jk i(e i )ω, i(e j )ω - 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω - i,j,k,l B ik , B jl i(e j ∧ e i )ω, i(e l ∧ e k )ω (12)
and the two last terms are zero for p = 1. Now, put

B + (ω) = i≤m i(e ⋆ i )ω ∧ B(e i , .
). The computation of the square of the norm of B + (ω) gives

|B + (ω)| 2 = 1 p! 1 ≤ i 1 , ..., ip ≤ m i, j (i(e ⋆ i )ω ∧ B(e i , .)) i 1 ...ip , i(e ⋆ j )ω ∧ B(e j , .) i 1 ...ip = 1 p! 1 ≤ i 1 , ..., ip ≤ m i, j, s, t (-1) s+t B iis , B jit ω ii 1 ... îs...ip ω ji 1 ... ît...ip
where the indices withˆare omitted. Then

|B + (ω)| 2 = 1 p! 1 ≤ i 1 , ..., ip ≤ m i, j, s
B iis , B jis ω ii 1 ... îs...ip ω ji 1 ... îs...ip

+ 1 p! 1 ≤ i 1 , ..., ip ≤ m i, j, s = t (-1) s+t B iis , B jit ω ii 1 ... îs...ip ω ji 1 ... ît...ip = 1 (p -1)! 1 ≤ i 1 , ..., i p-1 ≤ m i, j, k B ik , B jk ω ii 1 ...i p-1 ω ji 1 ...i p-1 - 1 p! 1 ≤ i 1 , ..., ip ≤ m i,
B ik , B jk i(e i )ω, i(e j )ω - 1 (p -2)! 1 ≤ i 1 ...i p-2 ≤ m i, j, k, l B ik , B jl ω ili 1 ...i p-2 ω jki 1 ...i p-2 = i,j,k B ik , B jk i(e i )ω, i(e j )ω + 1 (p -2)! 1 ≤ i 1 ...i p-2 ≤ m i, j, k, l B ik , B jl ω iji 1 ...i p-2 ω kli 1 ...i p-2 = i,j,k B ik , B jk i(e i )ω, i(e j )ω + i,j,k,l B ik , B jl i(e j ∧ e i )ω, i(e l ∧ e k )ω
Note that if M is Kaehlerian and ω is the Kaehler form then

|B + (ω)| 2 = |B + | 2 , where B + is the holomorphic part of B (i.e. B + (X, Y ) = 1 2 (B(X, Y ) + B(JX, JY )) where ω(X, Y ) = JX, Y ).
Now, combining the above relation with [START_REF] Sampson | Harmonic maps in Kähler geometry[END_REF], we obtain

|B + (ω)| 2 = m i,j B ij , H i(e i )ω, i(e j )ω + i,j (R φ ) ij i(e i )ω, i(e j )ω - 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω (13)
where the last term is zero if p

= 1. Putting X i 1 ...ip = i≤m ω ii 1 ...i p-1 e i and θ i 1 ...i p-2 = 1 2 1≤i,j≤m ω iji 1 ...i p-2 e ⋆ i ∧ e ⋆ j , we have i,j (R φ ) ij i(e i )ω, i(e j )ω = 1 (p -1)! i 1 ,...,i p-1 R φ (X i 1 ...ip , X i 1 ...ip ) ≤ (m -1) (p -1)! K 1 (φ(x)) i 1 ,...,i p-1 |X i 1 ...ip | 2 = p(m -1)K 1 (φ(x))|ω| 2 (14)
and

1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω = 2 (p -2)! i 1 ,...,i p-2 ρ(θ i 1 ...i p-2 , θ i 1 ...i p-2 ) ≥ 2 (p -2)! ρ 0 (φ(x)) i 1 ,...,i p-2 |θ i 1 ...i p-2 | 2 = 1 (p -2)! ρ 0 (φ(x)) i 1 ,...,ip ω 2 i 1 ...ip = p(p -1)ρ 0 (φ(x))|ω| 2 (15)
consequently, for all p ∈ {1, ..., m}, it follows from (13) and the fact that H = 0

|B + (ω)| 2 (x) ≤ p(m -1)K 1 (φ(x))|ω| 2 -p(p -1)ρ 0 (φ(x))|ω| 2
From this we conclude that if (p -1)ρ 0 (x) > (m -1)K 1 (x) holds for any x ∈ N , then there is no minimal immersion from (M m , g) into (N n , h).

Remark 3.3: In the theorem 3.1, the nonpositivity of the curvature operator of (N n , h) is not required. In ( [START_REF] Corlette | Archimedean superrigidity and hyperbolic geometry[END_REF]) Corlette proved (theorem 3.1) a similar result for harmonic maps φ from (M m , g) to (N n , h) (in fact he assumes that φ is a twisted harmonic map which is not necessary) by assuming only the nonpositivity of the curvature operator of (N n , h) without pinching condition. Indeed, under this hypothesis he proved that if (M m , g) is compact and has a parallel p-form then ∇ ⋆ (dφ ∧ ω) = 0 (here ∇ is the pullback of the Levi-Civita connection of T N ). Note that Corlette doesn't obtain a result of non existence. Moreover if φ is a minimal isometric immersion, then φ is a harmonic map and the theorem of Corlette can be proved without the compacity of M and a short computation shows that ∇ ⋆ (dφ ∧ ω) = -B + (ω).

Proof of theorem 3.2: From the relation (13) and the inequality (15) of the previous proof, we deduce that if φ is a minimal immersion, we have

|B + (ω)| 2 ≤ i,j (R φ ) ij i(e i )ω, i(e j )ω -p(p -1)ρ 0 (φ(x))|ω| 2
Since M is locally oriented, we can define locally the Hodge operator ⋆ and the m-p-form ⋆ω. But |B + (⋆ω)| 2 is independent on the choice of the orientability and is consequently globally defined. Then we have

|B + (ω)| 2 + |B + (⋆ω)| 2 ≤ i,j (R φ ) ij ( i(e i )ω, i(e j )ω + i(e i ) ⋆ ω, i(e j ) ⋆ ω ) -(p(p -1) + (m -p)(m -p -1)) ρ 0 (φ(x))|ω| 2 (16)
Now using the properties about inner and exterior products recalled in the first section, we have

i,j (R φ ) ij i(e i )ω, i(e j )ω + (R φ ) ij i(e i ) ⋆ ω, i(e j ) ⋆ ω = i,j (R φ ) ij i(e i )ω, i(e j )ω + (R φ ) ij e ⋆ i ∧ ω, e ⋆ j ∧ ω = i,j (R φ ) ij i(e i )ω, i(e j )ω + (R φ ) ij i(e j )(e ⋆ i ∧ ω), ω = i,j (R φ ) ij i(e i )ω, i(e j )ω + i (R φ ) ii |ω| 2 - i,j (R φ ) ij e ⋆ i ∧ i(e j )ω, ω = i (R φ ) ii |ω| 2 (17) A straightforward computation gives i (R φ ) ii ≤ Scal(φ(x)) -(n -m)(n + m -1)K 0 (φ(x))
and from ( 16) and ( 17) we deduce that for all

x ∈ M 0 ≤ Scal(φ(x)) -(n -m)(n + m -1)K 0 + (p(p -1) + (m -p)(m -p -1)) ρ 0 (φ(x))
Consequently if

Scal(x) < (n -m)(n + m -1)K 0 (x) + (p(p -1) + (m -p)(m -p -1)) ρ 0 (x)
for all x ∈ N , there is no minimal immersion from (M m , g) into (N n , h). Using the inequalities (2), we can easily see that the above condition implies

m(m -1) p(p -1) + (m -p)(m -p -1) K 0 (x) < ρ 0 (x) ≤ K 0 (x)
for all x ∈ N . And the theorem 3.2 is of interest only for K 0 < 0.

To finish this section, we study the particular case where the ambient space is the complex hyperbolic space CH n (c) with constant holomorphic curvature equal to c (c < 0). Let us recall that in this case the curvature tensor of CH n (c) has the expression

R(X, Y, Z, W ) = c 4 ( X ∧ Y, Z ∧ W + X ∧ Y, JZ ∧ JW + 2 X, JY Z, JW ) (18)
for all X, Y , Z, W ∈ Γ(T CH n (c)) . Here J denotes the complex structure of CH n (c).

For any isometric immersion φ of a Riemannian manifold (M m , g) into CH n (c) , we define the (1, 1)-tensor J φ on M by J φ X = i≤m Jdφ(X), dφ(e i ) e i , ∀X ∈ Γ(T M ) and for all orthonormal frame (e i ) 1≤i≤m . Recall that the immersion φ is said to be totally real if J φ ≡ 0. For this kind of immersions we have the Theorem 3.3 Let (M m , g) be an m-dimensional Riemannian manifold admitting a nontrivial parallel p-form (p ≥ 1). Then there is no minimal totally real immersion of (M m , g) into CH n (c).

Proof: Let φ be a minimal immersion of (M m , g) into CH n and assume that M has a non trivial parallel p-form ω. Then from (13) we have

|B + (ω)| 2 = i,j (R φ ) ij i(e i )ω, i(e j )ω - 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω
Now the conclusion follows from a straightforward computation. Using (18) the above equality becomes

|B + (ω)| 2 = c 4 p(m -p)|ω| 2 + 3 i,j≤m
J φ e i , J φ e j i(e i )ω, i(e j )ω

- i,j≤m i(e j ∧ e i )ω, i(J φ e j ∧ J φ e i )ω - i,j≤m i(e i ∧ J φ e i )ω, i(e j ∧ J φ e j )ω
where (e i ) 1≤i≤m is a local orthonormal frame.

4 Geometry of submanifolds with b p (M ) = 0

Let (M m , g) and (N n , h) be two Riemannian manifolds and assume that (M m , g) is compact. We use the same notations as in the previous sections. Moreover in this section, k(x) denotes the smallest eigenvalue at x of the Ricci curvature of (M m , g) and we put k 0 = min M (k(x)). On the other hand, if K 1 is bounded above, we will set K

1 max = max N (K 1 )
and ρ 1 max = max N (ρ 1 ). The first result is the following theorem Theorem 4.1 Let (M m , g) be a compact Riemannian manifold of dimension m ≥ 2 so that b p (M ) = 0 for some p ≥ 1. Then for any isometric immersion φ from (M m , g) into an n-dimensional Riemannian manifold (N n , h), there exists at least a point x ∈ M so that

m √ p p -1 p |B(x)||H(x)| ≥ k(x) - p -1 p ((m -1)K 1 + ρ 1 )(φ(x)) (19)
The following corollary is an immediate consequence of this theorem.

Corollary 4.1 Let (M m , g) be a compact Riemannian manifold of dimension m ≥ 2 minimally immersed in an n-dimensional Riemannian manifold (N n , h) (n > m). If ρ 1 is bounded above and if for an integer p so that 1 ≤ p ≤ m/2, we have

k 0 > p -1 p (m -1)K 1 max + ρ 1 max
then b q (M ) = 0 for q ∈ {1, ..., p}.

Remark 4.1: The inequality ( 19) is an equality at each point for the standard embedding of S p ( p/m) × S k 1 ( k 1 /m) × ... × S kq ( k q /m) into S m+r where p + k 1 + ... + k q = m with k i ≥ p (1 ≤ i ≤ q). For the case p = 2, (19) is also an equality at each point for the standard embedding from CP q with holomorphic curvature 2q/(q + 1) into S q 2 +2q . This theorem 4.1 is a generalization of a result obtained by El Soufi (see theorem 3.1 of [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF]) in the particular case p = 2.

To prove the theorem 4.1 we need the following proposition which gives an estimate of the term R p (ω), ω for any p-form ω. Proposition 4.1 Let (M m , g) be an m-dimensional compact Riemannian manifold isometrically immersed in an n-dimensional Riemannian manifold (N n , h). Then for any p ∈ {1, ..., m} and for any p-form ω of M , we have for all

x ∈ M R p (ω), ω ≥ p pk(x) -(p -1) (m -1)K 1 + ρ 1 (φ(x)) -m p -1 √ p |H(x)||B(x)| |ω| 2
Before proving this proposition, we introduce the following p-tensor associated to any p-form ω of M and the isometric immersion φ

B -(ω) = 1 (p -2)! i,i 1 ,...,i p-2 ≤m i(e i ∧ e i 1 ∧ ... ∧ e i p-2 )ω ∨ B(e i , .) ⊗ (e ⋆ i 1 ∧ ... ∧ e ⋆ i p-2 )
where (e i ) 1≤i≤m is an orthonormal frame at a point x ∈ M and ∨ denotes the symmetric product defined in the preliminaries (see ( 4)). It will be convenient to choose the norm

|B -(ω)| so that |B -(ω)| 2 = 1 (p -2)! jki 1 ...i p-2 ≤m B -(ω) jki 1 ...i p-2 2
Such a tensor has been introduced for the first time in [START_REF] Grosjean | A new Lichnerowicz-Obata estimate in the presence of a parallel p-form[END_REF] where the second fundamental form is replaced by the Hessian of a function. First note that if ω is a volume form at a point p of M where we have define the Hodge operator ⋆ so that ω = ⋆1, we deduce from (20) shown in the proof of the proposition 4.1, that

1 2n |B -(v g )| 2 = n -1 n ijk≤n B ik , B jk i(e i ) ⋆ 1, i(e j ) ⋆ 1 - 1 n ijkl≤n B ij , B kl i(e l ∧ e i ) ⋆ 1, i(e k ∧ e j ) ⋆ 1 = n -1 n ijk≤n B ik , B jk e ⋆ i , e ⋆ j - 1 n ijkl≤n B ij , B kl e ⋆ l ∧ e ⋆ i , e ⋆ k ∧ e ⋆ j = |B| 2 -n|H| 2 = |B -H ⊗ g| 2
In other words we obtain the square of the umbilicity tensor. Moreover if α denotes the Kaehler form of a Kaehlerian manifold, a straightforward calculation gives

|B -(α)| 2 = 4|B -| 2
where B -is the anti-holomorphic part of

B (i.e. B -(X, Y ) = 1 2 (B(X, Y ) -B(JX, JY )) where α(X, Y ) = JX, Y ).
On the other hand we will see in the proof of theorem 4.1 that B -(ω) is vanishing identically for the standard embedding of

S p ( p/m) × S k 1 ( k 1 /m) × ... × S kq ( k q /m) into S m+r where p + k 1 + ... + k q = m with k i ≥ p (1 ≤ i ≤ q).
Proof of the proposition 4.1: For p = 1 and from the relation [START_REF] Soufi | Immersions minimales et immersions pluriharmoniques entre variétés riemanniennes: résultats de non existence et de rigidité[END_REF], the equality of the proposition is obvious. Suppose now that p ≥ 2. Let x ∈ M and let (e i ) 1≤i≤m be a local orthonormal frame in a neighborhood of x. We have

(p -2)! 2 |B -(ω)| 2 = 1 2 i,
+ i,j,k,l,i 1 ,...,i p-2 i(e i ∧ e i 1 ∧ ... ∧ e i p-2 )ω k i(e j ∧ e i 1 ∧ ... ∧ e i p-2 )ω l B il , B jk = i,j,k,l,i 1 ...i p-2 B il , B jl ω iki 1 ...i p-2 ω jki 1 ...i p-2 + i,j,k,l,i 1 ,...,i p-2 B il , B jk ω iki 1 ...i p-2 ω jli 1 ...i p-2 = (p -1)! i,j,k B ik , B jk i(e i )ω, i(e j )ω -(p -2)! i,j,k,l B ij , B kl i(e l ∧ e i )ω, i(e k ∧ e j )ω
Finally, we have proved that

1 2 |B -(ω)| 2 = (p -1) i,j,k B ik , B jk i(e i )ω, i(e j )ω - i,j,k,l B ik , B jl i(e i ∧ e j )ω, i(e k ∧ e l )ω (20)
Now, combining this with the relations ( 10) and ( 11) and using the expression of R p (ω), ω (see ( 6)), we get

1 2 |B -(ω)| 2 = R p (ω), ω -p i,j Ric ij i(e i )ω, i(e j )ω + (p -1) i,j (R φ ) ij i(e i )ω, i(e j )ω + 1 2 i,j,k,l R ijkl i(e j ∧ e i )ω, i(e l ∧ e k )ω + m(p -1) i,j B ij , H i(e i )ω, i(e j )ω (21)
From the hypotheses on the curvature of N and by techniques already used in the proof of the theorem 3.1 (see ( 14) and ( 15)) we deduce that

1 2 |B -(ω)| 2 ≤ R p (ω), ω -p i,j Ric ij i(e i )ω, i(e j )ω + p(p -1) (m -1)K 1 (x) + ρ 1 (x) |ω| 2 + m(p -1) i,j B ij , H i(e i )ω, i(e j )ω (22)
Now, let us estimate the last term. For this, assume that at the point x ∈ M , (e i ) 1≤i≤m diagonalizes the symmetric tensor B(X, Y ), H . We have i,j

B ij , H i(e i )ω, i(e j )ω = 1 (p -1)! i,i 1 ,...,i p-1 B ii , H ω 2 ii 1 ...i p-1 = 1 p! i,i 1 ,...,i p-1 B ii , H + B i 1 i 1 , H + ... + B i p-1 i p-1 , H ω 2 ii 1 ...i p-1 ≤ 1 p! i,i 1 ,...,i p-1 |B ii | + |B i 1 i 1 | + ... + |B i p-1 i p-1 | |H|ω 2 ii 1 ...i p-1 ≤ √ p p! i,i 1 ,...,i p-1 |B||H|ω 2 ii 1 ...i p-1
Finally we have proved

i,j B ij , H i(e i )ω, i(e j )ω ≤ √ p|B||H||ω| 2 (23)
Since Ric ≥ kg, we have i,j

Ric ij i(e i )ω, i(e j )ω ≥ pk|ω| 2 , and we deduce from this, ( 22) and ( 23), the inequality of the proposition 4.1.

The proof of the theorem 4.1 is now an immediate consequence of the proposition 4.1

Proof of theorem 4.1: Since b p (M ) = 0, there exists a nontrivial p-form ω so that ∆ω = 0. And from the Weitzenböck formula (5), we deduce

M R p (ω), ω ≤ 0 (24)
Now, applying the estimate of the proposition 4.1 we get the desired inequality.

We can show a similar result to the theorem 4.1, with the scalar curvature Scal of (M m , g) instead of the Ricci curvature.

The theorem 4.2 and the inequality (26) was obtained by El Soufi (theorem 3.1 and theorem 3.2 of [START_REF] Soufi | Géométrie des sous-variétés admettant une structure kaehlérienne ou un second nombre de Betti non nul[END_REF]) in the particular case where m = 4 and p = 2. On the other hand, note that for p = m/2, the theorem 4.1 is not a consequence of the theorem 4.2.

For the same reasons as in the proof of theorem 3.2, we can choose locally an orientation on M and define locally the Hodge operator ⋆. But for all p-form ω of M , the quantity R m-p (⋆ω), ⋆ω is globally defined.

The theorem 4.2 is a consequence of the following proposition

Theorem 4 . 2 1 max + ρ 1 maxRemark 4 . 2 :

 421142 Let (M m , g) be a compact Riemannian manifold of dimension m ≥ 2 so that b p (M ) = 0 for a p ≥ 1. Then for any isometric immersion φ from (M m , g) into an n-dimensional Riemannian manifold (N n , h), there exists at least a point x ∈ M so thatm p -1 √ p + m -p -1 √ m -p |B(x)||H(x)| ≥ Scal(x) -(m -2)((m -1)K 1 + ρ 1 )(φ(x)) (25)We immediately deduce the following Corollary 4.2 Let (M m , g) be a compact Riemannian manifold of dimension m ≥ 2 minimally immersed into an n-dimensional Riemannian manifold (N n , h) (n > m). If ρ 1 is bounded above and if min M (Scal) > (m -2) (m -1)K then for any p ∈ {1, ..., m}, we have b p (M ) = 0. If p = m/2, we can improve (25) to obtain m(m -2)|H(x)| 2 ≥ Scal(x) -(m -2)((m -1)K 1 + ρ 1 )(φ(x))

Proposition 4 . 2 2 Remark 4 . 3 : 2 :

 422432 Let (M m , g) be an m-dimensional compact Riemannian manifold isometrically immersed in an n-dimensional Riemannian manifold (N n , h). Then for all p ∈ {1, ..., m -1} and for all p-form ω on (M m , g), we have for allx ∈ M R p (ω), ω + p m -p R m-p (⋆ω), ⋆ω ≥ p Scal(x) -(m -2) (m -1)K 1 + ρ 1 (φ(x)) -m p -1 √ p + m -p -1 √ m -p |H(x)||B(x)| |ω| If p = m/2, we can improve this inequality (see the proof of the proposition 4.2) to obtainR p (ω), ω + p m -p R m-p (⋆ω), ⋆ω ≥ p Scal(x) -(m -2) (m -1)K 1 + ρ 1 (φ(x)) -m(m -2)|H(x)| 2 |ω| 2 (27)And if for p ≥ 1, b p (M ) = 0, we get (26).Proof of the proposition 4.From the inequality (22) we obtain that for all p ≥ 1R p (ω), ω ≥ p i,j Ric ij i(e i )ω, i(e j )ω -p(p -1) (m -1)K 1 + ρ 1 (φ(x))|ω| 2 -m(p -1) i,j B ij , H i(e i )ω, i(e j )ω(28)Since ⋆ω is a (m -p)-form we have also R m-p (⋆ω), ⋆ω ≥ (m -p) i,j Ric ij i(e i ) ⋆ ω, i(e j ) ⋆ ω -(m -p)(m -p -1) (m -1)K 1 + ρ 1 (φ(x))|ω| 2

  j, s < t B iis , B jit ω iiti 1 ... îs... ît...ip ω jisi 1 ... îs... ît...ip B iis , B jit ω iiti 1 ... ît... îs...ip ω jisi 1 ... ît... îs...ip

	=
		i,j,k
	-	1 p!	1 ≤ i 1 , ..., ip ≤ m
			i, j, s > t

  j,i 1 ,...,i p-2 ∧ e i 1 ∧ ... ∧ e i p-2 )ω ∨ B(e i , .) kl , i(e j ∧ e i 1 ∧ ... ∧ e i p-2 )ω ∨ B(e j , .) kl = i,j,k,l,i 1 ,...,i p-2 i(e i ∧ e i 1 ∧ ... ∧ e i p-2 )ω k i(e j ∧ e i 1 ∧ ... ∧ e i p-2 )ω k B il , B jl

				B -(ω) i,j,i 1 ,...,i p-2	2 =
	=	1 2	i,j,k,l,i 1 ,...,i p-2	i(e i

Acknowledgements:

The author would like to express his gratefulness to Professor S.Ilias.

-m(m -p -1) i,j B ij , H i(e i ) ⋆ ω, i(e j ) ⋆ ω (29)

Multiplying (29) by p/(m -p), and summing the obtained inequality with (28), we find

By computations which are similar to (17) we get i,j

, and we show with the same arguments as in the proof of (30) that

and reporting this in (31), we obtain the inequality (27) of the remark 3.3. )

From the estimate (23), we deduce

now reporting these inequalities in (31), we find the inequality of the proposition.

The proof of the theorem is now immediate

Proof of theorem 4.2: Let ω be a harmonic p-form. Since the Hodge operator commutes with the Laplacian, then ⋆ω is a harmonic (m -p)-form, and we have

and the theorem follows from the proposition 4.2.

Remark 4.4:

We can improve all the results of this section by considering the particular case of submanifolds of the complex projective space CP n (c) (c > 0). We just need to compute in (21) the terms with the curvature tensor of the complex projective space. Then we obtain the same statements as previously by replacing (m -1)K


F Chautard

J L Baelde

F Bucaille

P Duneau

C Galard

J P Le Blay

J M Loyant

L Martina

M Ulrich

RADIOACTIVE BEAM DIAGNOSTICS STATUS AND DEVELOPMENT AT THE SPIRAL FACILITY

In 2001 the first radioactive beam will be accelerated in the CIME cyclotron of the SPIRAL project at GANIL. In order to be able to tune such low intensity beams in the cyclotron (down to few particles per second), a silicon and a scintillator detectors are mounted on probes. They measure the beam energy and the beam phase/RF in the cyclotron as a function of the radius. Such fragile diagnostics are meant to be used routinely from the control room by non-specialists in instrumentation and in presence of various beam intensities. Therefore, a program is developed to control and secure the acquisition procedure, the measurements and the isochronism correction. Additional detectors are installed at a fixed position in the ejection line before the experimental areas. Additionally, a diamond detector is foreseen to be installed in the machine to be tested in order to ease the CIME operation.

LOW ENERGY BEAM LINE

Because of the moderate resolution of the first mass selection out of the source (R=250) done by a dipole, it is necessary to have dedicated diagnostics [START_REF] Launé | Diagnostics for radioactive beams[END_REF] to detect the remaining ions after selection. This is done by a station connected to the low energy transport line by a switchyard. It will identify the ions by their radioactive properties and give the production rates of the source. This ion beam characterisation will be communicated to the cyclotron diagnostics to allow the tuning of the isochronism.

CYCLOTRON

General Layout

The tuning of the isochronism (beam phase/RF in function of the radius in the cyclotron φ(r)) in the CIME cyclotron required exotic diagnostics because of the low intensity beams accelerated.

Actually, different kind of detectors are implemented in the cyclotron (Fig. 1):

• A scintillator plastic BC418 from Eurisys (φ 15 mm) on a radial probe, • A silicon detector from Eurisys, 300 µm thick and 15x15 mm 2 on a second radial probe, • A BaF2 detector close to the vacuum chamber in a fixed position measuring the radiation emitted from the interception of the beam by a probe. • A scintillator and a Si detector, 300 µm thick and φ=15 mm at the exit of the cyclotron at a fixed position.

Figure 1: Detector positions in and around CIME

Acquisition and control

The analogic signals coming from the detectors (Fig. 2) are processed by a NIM electronic and converted to an ECL standard in order to be read by the VXI coders. A Fortran program does the control of the VXI processor from a standard VME station [START_REF] Piquet | The standard GANIL Data Acquisition System[END_REF]. The nuclear physicists, to plot the data in one or two dimension spectra, use a Ganil program, called VISUGAN. This program has some useful features already implemented as contour, filtering, rms calculation etc…. But, because of the future utilisation of the acquisition system by non-expert, a CP600, Cyclotrons and Their Applications 2001, Sixteenth International Conference, edited by F. Marti © 2001 American Institute of Physics 0-7354-0044-X/01/$18.00 graphical user interface (GUI) (Fig. 3) written in ADA [START_REF]Groupe informatique machine[END_REF] has been developed to combine:

• The equipment control (position and powering).

• The detector security control (beam flux limit).

• The VXI coders control.

• The calculation of the isochronism curve from a spectrum. • The plot via VISUGAN of the results.

• The data transfer to a tuning program. This application handles also the detector safety by inserting the pepperpots in the injection line for each measurement to reach less than 1000 particles per second on the detectors. Additionally, if during the measurement the counting increases above this rate, a fast kicker deflects the beam before the cyclotron entrance and a beam-stop is inserted that can be removed only by a manual action from the control room. This situation may happen because of the presence of the pre-tuning ion necessary to tune the machine. Its intensity should be high enough to be able to use the classical detector. Its presence in the machine after the exotic beam tuning and an equipment drift might bring it onto the exotic detectors and destroy it

In the case of the detectors mounted on probes, three different acquisition modes are possible:

• Fixed position: when the rate or the detector efficiency are too low. The acquisition time is increased as well as the statistic. • In flight (Fig. 4): there is an acquisition while the probe is moving. It is used when whether the counting is high enough or to have a first quick look at the spectrum before acquiring for statistic.

• Step-by-step (Fig. 5): The acquisition time for each step is entered, as well as the distance in millimetre between two measurements.

Another window is dedicated to the control of the VXI crate for initialisation, acquisition launch or acquisition stop.

An important feature is the possibility to choose an detector mode. When a mode is selected (scintillator probe, silicon probe, ejection scintillator, etc…) only events from the selected detectors are validated. Which eliminates noise or parasitic behaviour from the other detectors.

Finally and it is the main purpose of the program, this program should filter and extract the data of interest from a spectrum and fit a curve representing the isochronism law to be corrected. The equation of the curve will be sent to another existing GUI in order to calculate the correction current required in the eleven-isochronism coils and the main coil.

MEASUREMENTS

In the following, we show typical measurement examples done with the scintillator probe for a 16 O 3+ beam at 12 MHz RF frequency on harmonic 3. The ejection energy was 7.4 MeV/A. Figure 4

DEVELOPMENTS

A large effort from the Accelerator Division is done to develop new detectors adapted to the operation of CIME. The diamond detector [START_REF] Bergonzo | CVD Diamond for radiation detection[END_REF] with its good resistance to high particle flux and fast rise-time response makes a good candidate to be used by non-specialists from a control room. A collaboration is under way with the LIST from CEA/Saclay laboratory, France, to test the detector properties under different ion beams and see if it is well suited to our purpose.

CONCLUSIONS

Presently, the exotic diagnostics are used routinely and formation is given to spread their use to the whole Ganil operation group. At the end of last year a preliminary experiment with nuclear physicists was done to study beam acceleration in CIME with several ion species from Carbon to Xenon (Fig. 6). This fruitful collaboration allowed us to improve our technique. Then, new collaborations are foreseen and essential with

Figure 2 :

 2 Figure 2: Schematic architecture of the exotic diagnostic acquisition system

Figure 3 :

 3 Figure 3: Display of graphical user interface EXO_DIA.

  shows an in-flight acquisition. Due to the limited energy sensitivity of the plastic (1 or 2 MeV/A in function of the ion mass), the beam cannot be seen before a radius of 800 mm. Additionally, electric Control deflector is superimposed to the beam signal at ejection. Detector shielding should overcome this problem. As the isochronism curve is a smooth curve, only discrete points can be measured with longer acquisition time to increase the statistic. This is the measurement shown in Figure5.

Figure 4 :

 4 Figure 4: In-flight acquisition with a plastic detector.

Figure 5 :

 5 Figure 5: Plastic detector measurement with a step-bystep acquisition. An isochronism curve is superimposed.

Figure 6 :

 6 Figure 6: Energy measurement with a 300 µm silicon detector at a fixed position in the cyclotron during frequency variation with several ions accelerated. instrumentation specialists to enhance the development of such exotic equipments dedicated to accelerators.