Johannes Van Der

come


Seyed Ziaeddin Alborzi
email: seyed-ziaeddin.alborzi@inria.fr

Marie-Dominique Devignes
email: marie-domonique.devignes@loria.fr

David W Ritchie
email: dave.ritchie@inria.fr

Associating Gene Ontology Terms with Pfam Protein Domains

Keywords: Protein Structure, Protein Function, Gene Ontology, Content-Based Filtering

With the growing number of three-dimensional protein structures in the protein data bank (PDB), there is a need to annotate these structures at the domain level in order to relate protein structure to protein function. Thanks to the SIFTS database, many PDB chains are now cross-referenced with Pfam domains and Gene ontology (GO) terms. However, these annotations do not include any explicit relationship between individual Pfam domains and GO terms. Therefore, creating a direct mapping between GO terms and Pfam domains will provide a new and more detailed level of protein structure annotation. This article presents a novel content-based filtering method called GODM that can automatically infer associations between GO terms and Pfam domains directly from existing GO-chain/Pfam-chain associations from the SIFTS database and GO-sequence/Pfam-sequence associations from the UniProt databases. Overall, GODM finds a total of 20,318 non-redundant GO-Pfam associations with a F-measure of 0.98 with respect to the In-terPro database, which is treated here as a "Gold Standard". These associations could be used to annotate thousands of PDB chains or protein sequences for which their domain composition is known but which currently lack any GO annotation.

Introduction

Proteins carry out many important biological functions. At the molecular level, these functions are often performed by highly conserved regions called "domains". Currently, the Pfam database is one of the most widely used sequencebased classifications of protein domains and domain families [START_REF] Finn | Pfam: the protein families database[END_REF]. Protein domains may also be considered as building blocks which are combined in different ways in order to endow different proteins with different functions. A given Pfam domain might exist in several different proteins. It is widely accepted that protein domains often correspond to distinct and stable three-dimensional (3D) structures, and that there is often a close relationship between protein structure and protein function [START_REF] Berg | Protein structure and function[END_REF]. The Protein Data Bank (PDB) [START_REF] Bernstein | The protein data bank[END_REF][START_REF] Gutmanas | PDBe: protein data bank in europe[END_REF] contains more than 107,000 3D structures, that have been determined by X-ray crystallography or NMR spectroscopy. As well as sequence-based and structure-based classifications, proteins may also be classified according to their function. For example, the Gene Ontology (GO) [START_REF] Ashburner | Gene ontology: tool for the unification of biology[END_REF] organizes a controlled vocabulary describing the biological process (BP), molecular function (MF), and cellular component (CC) aspects of gene annotation. It provides an ontology of defined terms to unify the representation of the gene and protein roles in cells. The GO vocabulary is structured as a rooted Directed Acyclic Graph (rDAG) in which GO terms are nodes connected by different hierarchical relations. Each GO term within the gene ontology has a term name, a distinct alphanumeric identifier, and a namespace indicating to which ontology it belongs.

Although the GO is very useful, it does not generally provide a direct relationship between biological function and a (sequence-based) Pfam domain. Figure 1 illustrates the different kinds of relationships that can occur when considering GO-protein annotations at the domain level. Except for simple single-domain proteins where the mapping is obvious, it is generally not possible to compare and classify structure-function relationships at the domain level. An interesting exception is the dcGO database which provides multiple ontological annotations (Gene Ontology: GO, EC, pathways, phenotype, anatomy and disease ontologies) for protein domains [START_REF] Fang | dcGO: database of domain-centric ontologies on functions, phenotypes, diseases and more[END_REF]. In dcGO, an association between an ontology term and a domain is inferred from the principle that if a term tends to be attached to proteins in UniProtKB that contain a certain domain, then the term should be associated with that domain. For each Pfam domain, dcGO compares the number of Uniprot sequences containing that domain and annotated with a certain GO term to what could be obtained if association was random. The statistical significance of the association is then assessed using a hypergeometric distribution, followed by multiple hypotheses testing in terms of false discovery rate. Only significant associations are retained in the dcGO database. Nonetheless, we found that there are several GO-Pfam associations from manually curated data sources (e.g. InterPro) which are not present in dcGO. Moreover, based on our previous ECDomainMiner approach [START_REF] Alborzi | EC-PSI: associating enzyme commission numbers with Pfam domains[END_REF][START_REF] Alborzi | ECDomainMiner: discovering hidden associations between enzyme commission numbers and pfam domains[END_REF] to discover associations between EC numbers and protein domains, we found that there are many reliable EC-Pfam associations which are not covered by dcGO. Furthermore, there are thousands of protein structures in the PDB which lack GO annotations. If there is a direct association between protein domains and GO terms, these structures can be annotated through their associated domains. Based on our analysis, we estimated that dcGO associations can only annotate 43% of the unannotated PDB structures. Therefore, we were motivated to develop a more systematic approach, which we call "GODM" ("GO Domain Miner"), with the aim of discovering a much larger set of GO-domain associations than dcGO. GODM uses a "recommender-based" approach for finding direct associations between GO terms and Pfam domains. We recently developed a similar recommender-based approach called "ECDomainMiner" for assigning enzyme classification (EC) numbers to Pfam domains (manuscript accepted) [START_REF] Alborzi | ECDomainMiner: discovering hidden associations between enzyme commission numbers and pfam domains[END_REF]. Thus, the GODM approach described here represents a natural extension of our previously developed ECDomainMiner approach. Recommender systems are a subclass of information filtering system [START_REF] Hanani | Information filtering: Overview of issues, research and systems[END_REF][START_REF] Ricci | Introduction to recommender systems handbook[END_REF] which seek to predict a list of items that might be of interest to an on-line customer, and are divided into two main types. Collaborative filtering approaches make associations by calculating the similarity between activities of users [START_REF] Breese | Empirical analysis of predictive algorithms for collaborative filtering[END_REF][START_REF] Koren | Advances in collaborative filtering[END_REF]. In contrast, content-based filters predict associations between user profiles and description of items by identifying common attributes [START_REF] Ricci | Introduction to recommender systems handbook[END_REF][START_REF] Basu | Recommendation as classification: Using social and content-based information in recommendation[END_REF]. Here, we use content-based filtering to associate GO terms with Pfam domains from existing GO-chain and Pfam-chain associations from SIFTS [START_REF] Velankar | SIFTS: structure integration with function, taxonomy and sequences resource[END_REF], and GO-sequence and Pfam-sequence associations from Swis-sProt and TrEMBL. As well has handling simple one-to-one associations as in dcGO (Figure 1 part A), GODM can also resolve cases where multiple GO terms are associated with multi-domain chains (Figure 1 parts B, C, andD). While SwissProt and TrEMBL were originally developed separately, both databases have since been incorporated in the UniProt resource. SwissProt now represents a non-redundant, high quality, manually curated part of UniProt Knowledge Base (UniProtKB). In contrast, TrEMBL is an automatically annotated and unreviewed part of UniProtKB, and contains around 40 times more entries than SwissProt. In order to parameterise and evaluate our method, we use the InterPro database [START_REF] Mitchell | The InterPro protein families database: the classification resource after 15 years[END_REF] which contains a large number of manually curated GO-Pfam associations. We assess the performance of our approach against a "Gold Standard" dataset derived from InterPro, and we compare our results with the GO-Pfam associations available from the dcGO database. We also show how our database of more than 20,000 GO-Pfam associations for molecular function ontology can be exploited for automatic annotation purposes.

Methods

Data Preparation

Flat data files of SIFTS (July 2015), Uniprot (July 2015), and InterPro (version 53.0) were downloaded and parsed using in-house Python scripts. From the SIFTS data, associations between PDB chains and GO terms, and associations between PDB chains and Pfam domains were extracted in which each GO term is a leaf in the hierarchy of the Molecular Function ontology (GO-MF) and each Pfam refers either to a Pfam domain or a Pfam family (i.e. Pfam motifs and repeats were excluded). Associations between Uniprot sequence accession numbers (ANs) and GO terms from molecular function ontology, and AN-Pfam associations were then extracted from the SwissProt and TrEMBL sections of Uniprot to give two datasets of Swissprot associations and TrEMBL associations, respectively. Then, based on the evidence code of the GO term, associations in SwissProt and TrEMBL datasets were divided into two groups namely, associations for which GO terms were assigned in UniProtKB by manual curation, and Inferred from Electronic Annotation (IEA). These four datasets are subsequently called Swissprot, Swissprot-IEA, TrEMBL, and TrEMBL-IEA. Note that there are no evidence codes in the SIFTS.

To reduce bias due to the various numbers of identical sequences and sequences of chains in the five source datasets, all PDB chains and Uniprot sequences were grouped into clusters having identical sequences using the Uniref non-redundant cluster annotations [START_REF] Suzek | UniRef: domprehensive and non-redundant UniProt reference clusters[END_REF]. Each cluster was assigned a unique identifier (CID), and the source GO-chain and GO-AN associations were then mapped to the corresponding cluster in order to make five sets of GO-CID associations. A similar mapping was applied to the source Pfam-chain and Pfam-AN associations to make five sets of Pfam-CID associations.

For the InterPro reference data, we extracted a total of 1,561 GO-Pfam associations in which each GO term is a leaf node of the molecular function ontology and each Pfam refers to either a Pfam domain or a Pfam family. These associations were considered to be "true" associations. However, for training and filtering purposes, we also needed some examples of "false" associations. We therefore selected a set of the lowest-scoring GO-Pfam associations with the same size as InterPro dataset from the other datasets. These associations have to belong to at least two out of five datasets with no intersection with InterPro dataset. Because these associations have very little support in the data, we consider them to be "false" associations. Then, we randomly divided the InterPro dataset and our calculated "false" associations into two "Training" and "Test" subsets of the same size (each having half of the "true" and "false" associations). These two subsets were used for training and evaluation purposes respectively.

In the rest of this article, we will refer to the InterPro dataset as our "Gold Standard" dataset.

Finding GO-Pfam Associations by Content-Based Filtering

For each of the five datasets, all GO-CID relations are encoded in a binary (GO × CID) matrix, where a 1 represents the presence of a GO annotation and a 0 represents no annotation. This matrix is then row-normalised such that each row has unit magnitude when considered as a vector. Similarly, all CID-Pfam relations are encoded in a second binary (CID × Pfam) matrix which is column-normalised. Consequently, calculating the product of the two normalised matrices corresponds to calculating a matrix of cosine similarity scores between the rows of the first matrix and the columns of the second matrix. Thus, the product matrix represents an array of raw GO-Pfam association scores. Because we wish to draw upon the relations from all five input datasets, we combine the five scores to give a single normalized confidence score (CS):

CS go,d = i w i S i (go, d) i w i (1)
where i ∈ {SIFTS, Swissprot, Swissprot-IEA, TrEMBL, TrEMBL-IEA} enumerates the five datasets, w i are weight factors, to be determined, and where an individual association score, S i (go, d) is set to zero whenever there is no data for a given go and d. In order to calculate the weight factors, we calculated Receiver-Operator-Characteristic (ROC) curves [START_REF] Fawcett | An introduction to ROC analysis[END_REF] using the true associations from the Interpro Training set and all other associations as background associations. The weights were varied from 0.0 to 1.0 in steps of 0.1, and for each combination, associations were scored and ranked, and area under the curve (AUC) was calculated. Finally, we selected the combination of weights that gave the best area under the curve (AUC) of the ROC curve.

Defining a Confidence Score Threshold

Having determined the best weight for each data source, we next wished to determine a threshold for the confidence score. We scored and ranked the members of the training set of InterPro, and we divided the ranked list into two subsets according to a threshold value that was varied from 0.0 to 1.0 in steps of 0.01.

For each threshold value, we counted the number of true associations above the threshold, here called true positives (TPs), false associations above the threshold, false positives (FPs), false associations below the threshold, true negatives (TNs), and true associations below the threshold, false negatives (FNs). We then calculated the "F-measure" which is a harmonic mean of recall and precision using:

F = 2 × T P 2 × T P + F P + F N (2)
The score threshold that gave the best F-measure was confirmed by verifying that the F-measure calculated on the Test dataset is also very high. This threshold was thus selected as the best threshold to use for accepting predicted associations.

Hypergeometric Statistical Analysis

While the above procedure provides a systematic way to infer GO-Pfam associations, we wished to estimate the statistical significance, and thus the degree of confidence, that might be attached to those predictions. More specifically, we wished to calculate the probability, or "p-value", that a GO term and a Pfam domain could be found to be associated simply by chance. For example, it is natural to suppose such associations can be predicted at random if go or d are highly represented in the structure/sequence CIDs. In principle, in order to estimate the probability of getting our GO-Pfam associations by chance, one could generate random datasets by shuffling the relations between GO terms and CIDs on the one hand, and between Pfam domains and CIDs on the other hand. However, this is quite impractical given the very large numbers of CIDs, GO terms, and Pfam domains, and the complexity of the filtering procedure that would have to be repeated for each shuffled version of the dataset. Therefore, following [START_REF] Fang | dcGO: database of domain-centric ontologies on functions, phenotypes, diseases and more[END_REF], we assume that within each dataset (SIFTS, Swissprot, Swissprot-IEA, TrEMBL, or TrEMBL-IEA), the random hypothesis for the (go, d) association is represented by the hypergeometric distribution of the expected number of CIDs associated with both go and d.

Letting N denote the total number of CIDs, N d the number of CIDs related to the Pfam domain d, and N go the number of CIDs related to the GO term go, the hypergeometric probability distribution is given by

p(X go,d K go,d ) = min (N d ,Ngo) i=K go,d Ngo i N -Ngo N d -i N N d , (3)
where p(X go,d K go,d ) represents, in each dataset, the probability of having a number X go,d equal to or greater than the observed number K go,d of CIDs associated with both d and go. Traditionally, a p-value of less than 0.05 is taken to be statistically significant. However, because this test is applied to a large number of GO-Pfam associations, we apply a Bonferoni correction which takes into account the so-called family-wise error rate (FWER) [START_REF] Cui | Statistical tests for differential expression in cDNA microarray experiments[END_REF]. We therefore consider any p-value less than 0.05/T as denoting a statistically significant inferred GO-Pfam association in a dataset, with T the total number of tested GO-Pfam associations for that dataset.

Gold, Silver, And Bronze Associations

In order to differentiate associations based on their quality and reliability, our method categorizes associations into three classes of "Gold", "Silver", and "Bronze" using their calculated similarity scores and p-values. An association belongs to the Gold class if all its available p-values are statistically significant. The Silver class consists of associations for which the number of statistically significant p-values among the five datasets is greater than or equal to the number of statistically insignificant p-values (e.g. GO-Pfam is a Silver associations if its p-values are significant in SIFTS, SwissProt, and TrEMBL-IEA). The remaining associations are assigned to the Bronze class. An illustration of the whole procedure is shown in Figure 2.

Results

Our method takes as input five large datasets of GO-chain associations from SIFTS, and GO-sequence associations from SwissProt, SwissProt-IEA, TrEMBL and TrEMBL-IEA as well as five large datasets of Pfam-Chain and Pfam-sequence associations. These source datasets were merged to give a global dataset of 1, 161, 372 non-redundant GO-Pfam associations. The Training dataset consisting of 1,560 "true" and "false" GO-Pfam associations. The best ROC-plot AUC value of 0.99 was obtained with the weights w SIF T S = 10, w SwissP rot = 1, w SwissP rot-IEA = 10, w T rEM BL = 1, and w T rEM BL-IEA = 8. These weights clearly give a greater importance to the GO-Pfam associations from SIFTS and the IEA (Inferred from Electronic Annotation) section of SwissProt and TrEMBL compared to those derived from TrEMBL and the manually curated section of SwissProt.

In order to reduce the number of false associations predicted by our approach (and not just to simply optimise the overall AUC performance), various threshold values of the confidence score (using the above weights) were tested on the Training dataset using the F-measure (Section 2.3) with respect to the number of true and false associations having scores above or below the threshold. This gave an optimal threshold score of 0.01 for a maximum F-Measure of 0.99. Applying this threshold to the Test dataset yielded a recall value of 0.965 and a precision value of 1.0 to give a F-measure of 0.98. This threshold was then used to filter GO-Pfam associations from the merged dataset according to their confidence score. It is worth noting that if the ranked list of Test associations is evaluated with respect to the median rank (since the dataset contains equal numbers of true and false instances), our scoring function gives recall and precision values of 0.965, and thus a F-measure of only 0.965. This shows that using the chosen score threshold provides an objective way to achieve high recall with good precision (i.e. a low rate of false positive associations).

Analysis of Calculated GO-Pfam Associations

The summary of our calculated GO-Pfam associations are shown in Table 1. This table shows the numbers of GO-Pfam associations along with the numbers of distinct GO terms and Pfam entries involved in those associations for the five source datasets, our merged global dataset before and after filtering (the latter corresponding to our "GODM" GO-Pfam associations), and for the InterPro dataset of true associations. The overlap between these two last datasets is shown in the last line of the Overall, Table 1 shows that our approach yielded a total of 20, 318 GO-Pfam associations that include 1, 519 associations already present in InterPro. While this shows that our method finds 97.3% of the "correct" GO-Pfam associations in InterPro, it also shows that only 2.7% of the correct InterPro associations have confidence scores below our optimal score threshold of 0.01. This relatively high proportion of common associations reflects the fact that our method is designed to give relatively strong support (Confidence Score) to the correct associations in InterPro based on the five input sources.

Comparison Between our GODM and InterPro GO-Pfam Associations

Figure 3 (A) shows the average number of GO-Pfam associations per GO term and Pfam entry both for InterPro (shown in grey) and our calculated GODM dataset (in black). The ratio for our method is higher for GO terms (4.03 versus 2.64) and Pfam entries (3.3 versus 1.12), which reflects: i) a significant enrichment in the annotation of Pfam domains; and ii) participation of Pfam domains in different functions as either a single domain or a part of a complex. Figure 3 (B) shows the distribution of GO terms (in grey) and Pfam entries (in black) according to the number of associations they are involved in. More than 1,800 GO terms and 2,500 Pfam entries are involved in single associations, i.e. associated with a single Pfam domain and a single GO term respectively. Intersection of these single association sets yields a list of 135 one-to-one GO-Pfam associations. Nevertheless, the distribution also shows that our collection of associations rather favours multiple associations, thereby reflecting the complex many-to-many relationships that exist within the original datasets. Fig. 3. A: average number of GO-Pfam associations per GO terms and per Pfam entry for the InterPro (grey) and our calculated GODM (black) datasets. B: distribution of GO terms according to their numbers of associations with Pfam entries (grey). distribution of Pfam entries according to their numbers of associations with GO terms (black).

Comparing GODM and dcGO GO-Pfam Associations

In order to compare our results with dcGO [START_REF] Fang | dcGO: database of domain-centric ontologies on functions, phenotypes, diseases and more[END_REF], we extracted the Pfam2GO associations from the dcGO website (http://supfam.org/SUPERFAMILY/dcGO) where GO terms are leaves in the MF hierarchy of GO terms. This Pfam2GO dataset includes 3,086 GO-Pfam associations. Figure 4 shows that a total of 2,401 GO-Pfam associations are common to dcGO and our results (overlap B) while only 404 GO-Pfam associations are common between InterPro and dcGO (overlap C). Furthermore, this comparison shows that our GODM dataset contains 17,917 (20,318-2,401) additional GO-Pfam associations that are not available in the dcGO dataset. In a more detailed analysis, the overlap between the GODM and Pfam2GO datasets was studied with respect to our three quality classes. As summarized in the Table 2, the overlap between two datasets contains 1,621, 600, and 180 Gold, Silver, and Bronze associations, respectively.

Annotating PDB Chains with GO terms

Our analysis of the July 2015 release of the SIFTS database reveals that some 41% of PDB entries currently lack a leaf GO term annotation. More specifically, we found that a total of 48,409 PDB chains lacking GO annotations in SIFTS include at least one of the 6,154 Pfam domains present in our calculated GODM associations. For those chains, GODM finds 19,371, 7,176 and 12,530 Gold, Silver, and Bronze GO-Pfam associations, respectively, giving a total of 39,077 PDB chains that could benefit from the annotations inferred by GODM. Moreover, 153 PDB chains could benefit from non ambiguous one-to-one GO-Pfam associations.

To give an example, GODM finds a Gold association between PF03018 (Dirigent-like protein) and GO term GO:0042349 ("Guiding stereospecific syn-thesis activity"). Interestingly, the PF03018 domain is present in the PDB chain 4REV A ("Structure of the dirigent protein DRR206") which is not annotated by any GO term from the molecular function ontology. Consequently the GODM recommendation is to annotate the 4REV PDB entry with GO:0042349 term, which explicitly describes the possible function of this protein. Another example is PDB structure 2YRB, which is described only as "the solution structure of the first C2 domain from human KIAA1005 protein", and for which its previously assigned Pfam domain (PF11618) is annotated as a "protein of unknown function (DUF3250)". In this case, GODM finds a Gold association between PF11618 and GO:0031870 (thromboxane A2 receptor binding) thus indicating that this structure could be annotated with that GO term.

Conclusion

We have presented a systematic content-based filtering approach for assigning GO terms to protein domains and then categorizing those associations by a statistical method. This was achieved by first collecting existing annotations of protein chains or sequences, namely Pfam domain compositions on one hand and GO-MF leaf term annotations on the other.

We then applied the content-based filtering method to find a list of direct associations between GO-MF leaf terms and Pfam domains. Our approach is able to infer a total of 20,318 direct GO-Pfam associations. Thus, compared to the 1,561 manually curated GO-Pfam associations from InterPro database, our approach discovers over 13 times as many associations in a completely automatic way. We have also proposed some possible ways to further analyze the coverage of the our approach. We believe that the large numbers of GO-Pfam associations calculated using our approach can considerably contribute to enriching the annotations of PDB protein chains, and that this will facilitate a better understanding and exploitation of structure-function relationships at the protein domain level.

Fig. 1 .

 1 Fig. 1. A graphical representation of different situations of GO-Domain association in a protein sequence or structure.

Fig. 2 .

 2 Fig. 2. A schematic overview of the GODM procedure.

Fig. 4 .

 4 Fig. 4. Venn diagram showing the intersection between Pfam2GO (3,086 associations) from dcGO, our GODM associations (20,318 associations), and manually curated associations (1,561 associations) from InterPro. Region A (1,519 associations) is the overlap between our result and InterPro associations. Region B (2,401 associations) is the common associations between our result and Pfam2GO. Region C (404 associations) is the overlap between Pfam2GO and InterPro associations.

Table 1 .

 1 table. Statistics on the given and our result GO-Pfam associations.

	Dataset	GO-Pfam associations GO terms Pfam entries
	SIFTS	10,064	2,763	3,370
	SwissProt	22,435	4,220	4,669
	SwissProt-IEA	28,982	3,228	4,469
	TrEMBL	22,031	2,766	3,613
	TrEMBL-IEA	1,136,711	4,254	9,342
	Merged	1,161,372	5,510	9,929
	InterPro	1,561	591	1,390
	Our Associations (GODM)	20, 318	5, 047	6, 154
	Common with InterPro	1,519	586	1,362

Table 2 .

 2 Overlap between associations in GODM classes, Pfam2GO of dcGO, and InterPro.

	Dataset	GODM	Overlap with
			Pfam2GO	InterPro
	Gold	9,771	1,621	922
	Silver	4,280	600	455
	Bronze	6,267	180	72
	Total	20,318	2401	1,519

Acknowledgments

This project is funded by Agence Nationale de la Recherche (grant number ANR-11-MONU-006-02), the Institut National de Recherche en Informatique et Automatique, and Région Lorraine.


Philippe Jouan
email: philippe.jouan@univ-rouen.fr

Zsigmond † Guilherme

Victor Ayala
email: vayala@ucn.cl

Guilherme Zsigmond

Isometries of almost-Riemannian structures on Lie groups

Keywords: Lie groups, Linear vector fields, Almost-Riemannian geometry, Isometries

published or not. The documents may come

Introduction

This paper is devoted to isometries of Almost-Riemannian structures on Lie groups. The purpose is to classify these structures, to find geometric invariants, and to determine their groups of isometries.

An almost-Riemannian structure (ARS in short) on an n-dimensional differential manifold can be defined, at least locally, by a set of n vector fields, considered as an orthonormal frame, that degenerates on some singular set. This geometry goes back to [START_REF] Grushin | A certain class of hypoelliptic operators[END_REF] and [START_REF] Takasu | Generalized Riemannian Geometry I[END_REF]. It appears as a part of sub-Riemannian geometry, and has aroused some interest, as shown by the recent papers [START_REF] Agrachev | A Gauss-Bonnet like formula on two-dimensional almost-Riemannian manifolds[END_REF], [START_REF] Agrachev | Two dimensional almost-Riemannian structures with tangency points[END_REF], [START_REF] Bonnard | Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF], [START_REF] Bonnard | The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry[END_REF], [START_REF] Boscain | Mason Local properties of almost-Riemannian structures in dimension 3[END_REF], [START_REF] Boscain | Lipschitz classification of almost-Riemannian distances on compact oriented surfaces[END_REF], [START_REF] Boscain | Normal forms and invariants for 2-dimensional almost-Riemannian structures[END_REF].

On an n-dimensional connected Lie group the simplest ARSs are defined by a set of n -1 left-invariant vector fields and one linear vector field, the rank of which is equal to n on a proper open and dense subset and that satisfy the rank condition (a vector field on a Lie group is linear if its flow is a one parameter group of automorphisms, see Section 2, and [START_REF] San Martin | Controllability properties of a class of control systems on Lie Groups[END_REF], [START_REF] Ayala | Tirao Linear control systems on Lie groups and Controllability[END_REF], [START_REF] Dath | Jouan Controllability of linear systems on low dimensional nilpotent and solvable Lie groups[END_REF], [START_REF] Ph | Controllability of linear system on Lie groups[END_REF], [START_REF] Ph | Jouan Invariant measures and controllability of finite systems on compact manifolds[END_REF] about linear systems on Lie groups).

These ARSs, among which we find the famous Grushin plane on the Abelian Lie group R 2 , has been studied in [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF]. Among the results of this paper there is a study of the singular locus, that is the set of points where the vector fields fail to be independent. It is an analytic set, but not a subgroup, not even a submanifold, in general. However sufficient conditions for the singular locus to be a submanifold or a subgroup were exhibited and are recalled in Section 2. This locus is very important in what concern the structure of ARSs, in particular in view of a classification. Another important geometric locus is the set of singularities of the linear field. It is always a subgroup.

In this paper we deal with smooth isometries, i.e. diffeomorphisms that respect the Euclidean metric of the tangent space at each point. First of all we show that such an isometry should preserve the singular locus and the group of singularities of the linear field. The main consequence is that the group of isometries of an ARS does not act transitively on G. Another consequence is that a left translation L g is an isometry if and only if g belongs to the set of singularities of the linear field.

Then we prove that the isometries preserve the left-invariant distribution generated by the n -1 left-invariant vector fields, and also the linear field (up to the sign), see Theorem 2.

These constraints are rather strong. Consider for instance ARSs on the Heisenberg group, in the case where the distribution generated by the left-invariant vector fields is a subalgebra. Then the group of isometries is generically reduced to the identity.

Kivioja and Le Donne proved in [START_REF] Kivioja | Le Donne Isometries of nilpotent metric groups[END_REF] that the isometries of left-invariant metrics on nilpotent groups are affine, that is composed of an automorphism and a left translation. Their result cannot be directly applied here since the metric is not left-invariant. It can however be adapted, and it is shown in Theorem 4 that the isometries of ARSs on nilpotent Lie groups are affine.

The paper is organized as follows. In Section 2 the basic definitions and notations are stated, together with some useful results.

Section 3 is devoted to the general theorems quoted above.

In Sections 4 and 5 the ARSs of the 2D affine group and the 3D Heisenberg group are completely classified. We begin by a classification by isometries, up to a global rescaling of the metric that does not modify the geometry. In order to reduce the number of parameters we then identify ARSs that differ only by the left-invariant metric on the left-invariant distribution. The typical ARSs that we obtain are characterized by the following geometric invariants: the singular locus, the set of singularities of the linear field, the tangency points and the eigenvalues of the derivation associated to the linear field.

Basic definitions

Linear vector fields

In this section the definition of linear vector fields and some of their properties are recalled. More details can found in [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF].

Let G be a connected Lie group and g its Lie algebra (the set of left-invariant vector fields, identified with the tangent space at the identity). A vector field on G is said to be linear if its flow is a one-parameter group of automorphisms. Notice that a linear vector field is consequently analytic and complete. The flow of a linear vector field X will be denoted by (ϕ t ) t∈R .

The following characterization will be useful in the sequel.

Characterization of linear vector fields

A vector field X on a connected Lie group G is linear if and only if X belongs to the normalizer of g in the algebra V ω (G) of analytic vector fields of G and verifies X (e) = 0, that is

∀Y ∈ g [X , Y ] ∈ g and X (e) = 0. (1)
According to (1) one can associate to a given linear vector field X the derivation D of g defined by:

∀Y ∈ g DY = -[X , Y ],
that is D = -ad(X ). The minus sign in this definition comes from the formula [Ax, b] = -Ab in R n . It also enables to avoid a minus sign in the useful formula:

∀Y ∈ g, ∀t ∈ R ϕ t (exp Y ) = exp(e tD Y ). (2)
An affine vector field is an element of the normalizer N of g in V ω (G), that is

N = norm V ω (G) g = {F ∈ V ω (G); ∀Y ∈ g, [F, Y ] ∈ g},
so that an affine vector field is linear if and only if it vanishes at the identity. It can be shown (see [START_REF] Ayala | Tirao Linear control systems on Lie groups and Controllability[END_REF] or [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF]) that an affine vector field can be uniquely decomposed into a sum F = X + Z where X is linear and Z is right-invariant.

Almost-Riemannian structures

For all that concern general sub-Riemannian geometry, including almost-Riemannian one, the reader is referred to [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian Geometry[END_REF]. About Almost-Riemannian structures on Lie groups more details can be found in [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF].

Definition 1 An almost-Riemannian structure on an n-dimensional Lie group G is defined by a set of n vector fields {X , Y 1 , . . . , Y n-1 } where (i) X is linear; (ii) Y 1 , . . . , Y n-1 are left-invariant; (iii) n = dim G and the rank of X , Y 1 , . . . , Y n-1 is full on a nonempty subset of G; (iv) the set {X , Y 1 , . . . , Y n-1 } satisfies the rank condition.
The metric is defined by declaring the frame {X , Y 1 , . . . , Y n-1 } to be orthonormal.

Equivalent definition. An ARS on a Lie group can as well be defined by an (n-1)-dimensional left-invariant distribution ∆ (that is ∆ = Span{Y 1 , . . . , Y n-1 }), a left-invariant Euclidean metric on ∆ and a linear vector field X assumed to satisfy the conditions (iii) and (iv) of Definition 1. The metric of the ARS is then defined by declaring X unitary and orthogonal to ∆.

Necessary conditions for the rank condition

First notice that if [∆, ∆] ⊆ ∆ and D(∆) ⊆ ∆, then the Lie algebra generated by X , Y 1 , . . . , Y n-1 is equal to RX ⊕ ∆. But the rank of that Lie algebra is not full at the identity e. Consequently the rank condition implies that at least one of the following conditions hold:

(i) [∆, ∆] ∆ (ii) D(∆) ∆
In both cases, the full rank is obtained after one step.

Singular locus

The set where the rank of X , Y 1 , . . . , Y n-1 is not full will be referred to as the singular locus and denoted by Z. It is an analytic subset of G. By Assumption (iii) it is not equal to G, and by analycity its interior is empty. On the other hand X (e) = 0 and it cannot be empty. Finally G \ Z is an open, dense and proper subset of G.

The set of singularities of the linear field X will be denoted by Z X = {g ∈ G; X (g) = 0}. It is a subgroup of G, the Lie algebra of which is ker(D), where D is the derivation associated to X . It is included in Z but different in general.

The points of G \ Z will be called the Riemannian points and the ones of Z the singular points.

Norms and isometries

The almost-Riemannian norm on T g G is defined by:

For X ∈ T g G, X = min    v 2 + n 1 u 2 i ; vX g + u 1 Y 1 (g) + • • • + u n-1 Y n-1 (g) = X    .
It is infinite if the point g belongs to the singular locus and X does not belong to ∆ g .

Definition 2 Let (Σ) and (Σ ′ ) be two ARSs on the Lie group G. An isometry Φ from (Σ) onto (Σ ′ ) is a diffeomorphism of G that respects the norms, that is:

∀g ∈ G, ∀X ∈ T g G T g Φ.X Σ ′ = X Σ .
where . Σ (resp. . Σ ′ ) stands for the norm associated to (Σ) in T g G (resp. to (Σ ′ ) in T Φ(g) G).

Summary of some results

The following results are proved in [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF].

Theorem 1 of [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF] If ∆ is a subalgebra of g then the singular locus Z is an analytic, embedded, codimension one submanifold of G. Its tangent space at the identity is D -1 ∆.

Theorem 2 of [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF] If the Lie algebra g is solvable and ∆ is a subalgebra of g, then the singular locus Z is a codimension one subgroup of G whose Lie algebra is z = D -1 ∆.

Notations

In the sequel the following notations will be used:

1. Let Y ∈ g. Then Y g stands for T L g .Y , where L g is the left translation by g, and T L g its differential. The Euclidean norm in T g G is denoted by . g , or simply by . if no confusion is possible. Notice that if g ∈ Z then . g is only defined on ∆ g .

2.

To the linear vector field X we associate F (g) = T L g -1 .X g ∈ g. In some expressions we also write F g for F (g) when it is lighter, for instance in ad(F g ).

Finally the following formula, proven in [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF], will be used in some proofs.

T g F = (D + ad(F g )) • T L g -1 . (3)
Notice that here F g is an element of g.

General Theorems about Isometries of ARSs

In this section (Σ) and (Σ ′ ) stand for two simple ARSs on the same Lie group G. An object related to (Σ ′ ) will be denoted by the same symbol as the analogous object related to (Σ) but with a prime, that is, D ′ , (ϕ ′ t ) t∈R , and so on.

Translations

The purpose of this subsection is to characterize the left translations that are isometries and to show that there exists an isometry from (Σ) onto (Σ ′ ) if and only if there exists one such isometry that preserves the identity.

Proposition 1 Let Φ be an isometry from (Σ) onto (Σ ′ ) . Then Φ sends the singular locus of (Σ) onto the one of (Σ ′ ) and the set of fixed points of X onto the one of X ′ , that is:

Φ(Z) = Z ′ and Φ(Z X ) = Z ′ X ′ .
In particular X ′ Φ(e) = 0.

Proof. Since the singular locus is the set of points of G where the rank of Span{X , Y 1 , . . . , Y n-1 } is not full, it is clear that Φ(Z) = Z ′ . We are left to prove that Φ(Z X ) = Z ′ X ′ .

Let V ∈ ∆ e , V = 0. It can be written in an unique way as V = n-1 i=1 a i Y i and we may assume without lost of generality that V e = n-1 i=1

a 2 i = 1. Let g ∈ Z X and V g = T L g .V . Since X (g) = 0 the vector V g writes V g = n-1 i=1 a i Y i (g) in an
unique way, and consequently

V g g = 1 = V e . Let now g ∈ Z \ Z X . At such a point X (g) = n-1 i=1 b i Y i (g) = 0. Let b = n-1 i=1 b 2 i > 0 and let us choose the particular V = n-1 i=1 a i Y i where a i = b -1 b i . Then V g = T L g .V writes V g = n-1 i=1 a i Y i (g)
but also as

V g = vX g + n-1 i=1 u i Y i (g) with vb i + u i = a i , i = 1, . . . , n -1.
The square of the norm of

V g is inf{v 2 + n-1 i=1 u 2 i ; vb i + u i = a i for i = 1, . . . , n -1}. A straight-
forward computation shows that the minimum is attained for v = b 1 + b 2 and consequently that

V g 2 g = 1 1 + b 2 < 1 though V e = 1.
This shows that the function g → T L g .V is not continuous at g. Therefore the sub-Riemannian metric is continuous at points of Z X (and at Riemannian points of course) but discontinuous at the points of Z \ Z X . This implies that the image of the set Z \ Z X by the isometry Φ should be Z ′ \ Z ′ X ′ , which is the desired conclusion.

Proposition 2 Let g ∈ G. The left translation L g is an isometry of (Σ) if and only if g belongs to the set Z X of fixed points of X .

Proof.

According to Proposition 1 the condition is necessary. Conversely let g ∈ Z X . Since Y 1 , . . . , Y n-1 are left-invariant we have (L g ) * Y i = Y i for i = 1, . . . , n -1. On the other hand (L g ) * X = X + Z for some right-invariant vector field Z. Indeed

∀Y ∈ g [Y, X ] = (L g ) * [Y, X ] = [(L g ) * Y, (L g ) * X ] = [Y, (L g ) * X ].
This proves that (L g ) * X is an affine vector field whose adjoint action on g is the same as the one of X . According to [START_REF] Ph | Jouan Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces[END_REF] it is equal to X up to a right-invariant vector field Z.

Moreover (X + Z)(g) = T L g .X e = 0. But X g = 0 and necessarily Z = 0. This proves that (L g ) * X = X , and together with (L g ) * Y i = Y i for i = 1, . . . , n -1, that L g is an isometry.

Theorem 1 The ARSs (Σ) and (Σ ′ ) are isometric if and only if there exists an isometry Φ from (Σ) onto (Σ ′ ) such that Φ(e) = e.

Proof.

Let Ψ be an isometry from (Σ) onto (Σ ′ ) . Firstly Ψ(e) ∈ Z ′ X ′ according to Proposition 1. According to Proposition 2 the left translation L Ψ(e) -1 is an isometry of (Σ ′ ) . Then Φ = L Ψ(e) -1 • Ψ suits.

Preservation of the distribution

We recall the following notations: if

Y ∈ T e G and g ∈ G then Y g stands for T L g .Y ∈ T g G.
The key point of this subsection is the following lemma that characterizes the elements of ∆ g (when g does not belong to the singular locus): they are the vectors whose norm is invariant under (small) left translations.

Lemma 1 Let Σ be an ARS on the Lie group G.

There exists an open and dense subset of G\Z, hence of G, of points g that satisfy:

∆ g = {Y g ; Y h h = Y g g in a neighborhood of g}. Proof. Let g ∈ G \ Z and Y ∈ T e G. If Y = n-1 i=1 a i Y i ∈ ∆, then Y h writes uniquely as Y h = n-1 i=1 a i Y i (h)
and Y h h is constant in a neighborhood of g. We have to show that this is no longer true if Y / ∈ ∆. Since the singular locus is an analytic set, we can find X ∈ T e G such that exp(tX) / ∈ Z for t in some open interval (0, τ ). Recall that F stands for the mapping from G to T e G defined by F (g) = T L g -1 .X g . The previous condition is equivalent to F (exp(tX)) / ∈ ∆ for t ∈ (0, τ ), or better to: ω, F (exp(tX)) = 0 for t ∈ (0, τ )

where ω stands for a nonvanishing one-form on g orthogonal to ∆. But Condition (4) together with ω, F (e) = 0 imply d dt ω, F (exp(tX)) = 0 on (0, τ ) (except may be at isolated points). Consequently the set

Ω = {g ∈ G; g / ∈ Z and d dt |t=0 ω, F (g exp(tX)) = 0}
is not empty. This set being defined by analytic conditions is open and dense in G.

Let g ∈ Ω, let us consider the path γ(t) = F (g exp(tX)) in T e G and let Y = γ(0) = F (g). Since g / ∈ Z the vector F (g) does not belong to ∆ e , the n-uple (F (g), Y 1 , . . . , Y n-1
) is a basis of T e G, and the derivative γ ′ (0) can be written as γ

′ (0) = vF (g) + n-1 i=1 u i Y i = vY + n-1 i=1 u i Y i . Then we have γ(t) = Y + tγ ′ (0) + O(t 2 )) hence Y = γ(t) -tγ ′ (0) + O(t 2 ) = γ(t) -t(vγ(0) + n-1 i=1 u i Y i ) + O(t 2 ) and (1 + tv)Y = γ(t) -t n-1 i=1 u i Y i + O(t 2 ).
Let us denote by . t the Euclidean norm in T e G defined by the moving orthonormal frame (γ(t), Y 1 , . . . , Y n-1 ). We have

(1 + tv) 2 Y 2 t = 1 + O(t 2 ) and Y 2 t = 1 (1 + tv) 2 + O(t 2 ).
From that we obtain

d dt |t=0 Y 2 t = -2v.
Since g ∈ Ω the vector γ ′ (0) does not belong to ∆ and v = 0. Moreover we have obviously Y γ(t) γ(t) = Y t , which proves that in any neighborhood of g, Y h h is not constant. The same happens for any Z that does not belong to ∆ because it is the sum of an element of ∆ and of αY for some nonzero real number α.

Theorem 2 Let Φ be an isometry from (Σ) onto (Σ ′ ) that preserves the identity. Then:

1. Its tangent mapping Φ * sends ∆ on ∆ ′ , that is, T g Φ.∆ g = ∆ Φ(g) for all g ∈ G. 2. Either Φ * X = X ′ , and T e Φ • D = D ′ • T e Φ or Φ * X = -X ′ , and T e Φ • D = -D ′ • T e Φ.
Proof. According to Lemma 1 there is an open and dense subset Ω of points g of G \ Z where the distribution is characterized by:

∆ g = {Y g ; Y h h = Y g g in a neighborhood of g}.
This characterization should be preserved by the isometry Φ hence T g Φ(∆ g ) = ∆ ′ Φ(g) . The density of Ω implies that this equality actually holds at all points.

Let now g ∈ G\Z. The image by Φ of the orthonormal frame (X (g), Y 1 (g), . . . , Y n-1 (g)) is the orthonormal frame (T g Φ.X (g), T g Φ.Y 1 (g), . . . , T g Φ.Y n-1 (g)). But the fact that T g Φ.∆ g = ∆ ′ Φ(g) implies that T g Φ.X (g) is an unitary vector orthogonal to ∆ ′ Φ(g) , that is,

T g Φ.X (g) = ±X ′ (Φ(g)). (5)
In this equality the sign is constant on any connected subset of G where X does not vanish. On the other hand the set Z X of singularities of X is a subgroup of G whose Lie algebra is ker(D). The set G \ Z X can be disconnected only if the dimension of ker(D) is n -1. In that case let Y be any element of T e G that does not belong to ker(D). Then exp(tY ) does not belong to Z X , for t in some interval (0, T ), with T > 0. The sign of the equality T exp(tY ) Φ.X (exp(tY )) = ±X ′ (Φ(exp(tY ))) being constant for t ∈ (0, T ), we denote it by ǫ(Y ) = ±1. We push this equality to T e G and we derivate it at t = 0. On the one hand we get (here ϕ t stands for the flow of X and we use Formula ( 2)):

d dt |t=0 T L (Φ(exp(tY ))) -1 .T Φ(exp(tY )) .X (Φ(exp(tY ))) = d dt |t=0 d ds |s=0 (Φ(exp(tY ))) -1 Φ(ϕ s (exp(tY ))) = d dt |t=0 d ds |s=0 (Φ(exp(tY ))) -1 Φ(exp(e sD tY )) = d ds |s=0 (-T e Φ.Y + T e Φe sD Y ) = T e ΦDY,
and on the other hand:

d dt |t=0 T L (Φ(exp(tY ))) -1 .X ′ Φ(exp(tY ) = d dt |t=0 F ′ (Φ(exp(tY )) = D ′ T e Φ.Y
according to Formula (3) recalled in Section 2.

Finally we have obtained T e ΦDY = ǫ(Y )D ′ T e Φ.Y . By linearity we have also T e ΦD(-Y ) = ǫ(Y )D ′ T e Φ.(-Y ), and consequently T exp(-tY ) Φ.X (exp(tY )) = ǫ(Y )X ′ (Φ(exp(-tY ))) for t small enough. But exp(tY ) and exp(-tY ) are located in different connected components of G \ Z X (if different connected components exist), and this proves that the sign is constant in Formula (5).

Remark. Since the replacement of X ′ by -X ′ does not modify the ARS (Σ ′ ) we can always assume that Φ * X = X ′ .

The tangent mapping of an isometry

Theorem 3 Let Φ 1 and Φ 2 be two isometries from (Σ) onto (Σ ′ ) that preserve the identity.

If

T e Φ 1 = T e Φ 2 then Φ 1 = Φ 2 .
Before proving the theorem let us recall that the normal Hamiltonian of an ARS is

H = 1 2 λ g , X (g) 2 + 1 2 n-1 i=1 λ g , Y i (g) 2 ,
where λ g ∈ T * g G. At Riemannian points the geodesics are the projections of the integral curves of the Hamiltonian vector field associated to H (see [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian Geometry[END_REF] for more information).

Proof. The diffeomorphism Φ -1 2 • Φ 1 being an isometry of (Σ) , it is sufficient to prove that an isometry of (Σ) that preserves the identity e and whose tangent mapping at e is the identity of T e G is itself the identity mapping of G.

So let Φ be such an isometry of (Σ) .

In order to simplify the notation let Ψ be the lift of Φ to the cotangent space, that is

Ψ(g, λ g ) = (Φ(g), λ g • T Φ(g) Φ -1 ).
It is a classical fact that Ψ is a symplectomorphism. According to Proposition 4.51 of [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian Geometry[END_REF] we have

Ψ * -→ H = -----→ H • Ψ -1
where H stands for the normal sub-Riemannian Hamiltonian and -→ H for the associated Hamiltonian vector field.

Let g be a Riemannian point of G. A straightforward computation shows that:

H • Ψ -1 (g, λ g ) = 1 2 λ g , T Φ -1 (g) Φ.X (Φ -1 (g)) 2 + 1 2 n-1 i=1 λ g , T Φ -1 (g) Φ.Y i (Φ -1 (g)) 2 . ( 6
)
Since Φ is an isometry,

(T Φ -1 (g) Φ.X (Φ -1 (g)), T Φ -1 (g) Φ.Y 1 (Φ -1 (g)), . . . , T Φ -1 (g) Φ.Y n-1 (Φ -1 (g)))
is an orthonormal basis of T g G and ( 6) is half the sum of the squares of the coordinates of λ g in the dual basis. It is a standard fact of linear algebra that this is invariant by isometry. Consequently

H • Ψ -1 (g, λ g ) = 1 2 λ g , X (g) 2 + 1 2 n-1 i=1 λ g , Y i (g) 2 = H(g, λ g ).
The set of Riemannian points being dense in G, this implies

Ψ * -→ H = -----→ H • Ψ -1 = -→ H. Let (g(t)
, λ(t)) be a normal extremal with initial conditions g(0) = e, and λ(0) = λ g ∈ T * e G. Then Ψ(g(t), λ(t)) is also a normal extremal with initial conditions e and (T e Φ -1 ) * λ g . Since T e Φ = Id we get Φ(g(t)) = g(t) for all t for which g(t) is defined. This is enough to prove that Φ = Id.

Nilpotent groups

Theorem 4 If the group G is nilpotent then the isometries of ARSs of G that preserve the identity are automorphisms.

Proof. Let (Σ) and (Σ ′ ) be two ARSs on G, and Φ an isometry from (Σ) onto (Σ ′ ) that preserves the identity.

(i) ∆ and ∆ ′ are not subalgebras. In that case ∆ and ∆ ′ together with their left-invariant metrics define left-invariant sub-Riemannian metrics on G. Since T g Φ is an isometry from ∆ g onto ∆ Φ(g) for all g ∈ G, the diffeomorphism Φ is an isometry of these left-invariant structures. According to the theorem of Kivioja-Le Donne ( [START_REF] Kivioja | Le Donne Isometries of nilpotent metric groups[END_REF]), Φ is an automorphism.

(ii) ∆ and ∆ ′ are subalgebras. In that case ∆ and ∆ ′ define left-invariant integrable distributions on G. If we denote by H (resp. by H ′ ) the connected Lie subgroup generated by ∆ (resp. by ∆ ′ ) then the leaves of the induced foliations are the cosets gH (resp. the cosets gH ′ ).

Let g ∈ G, and let M = Φ(gH) be the image of the coset gH by Φ. Since Φ is a diffeomorphism, M is a submanifold of G, and for any h ∈ H the tangent space at the point Φ(gh) is

T gh Φ.∆ gh = ∆ ′ Φ(gh)
. Consequently M is the leaf through the point Φ(g) of the distribution ∆ ′ , hence equal to Φ(g)H ′ , and Φ is an isometry from gH onto Φ(g)H ′ (because T g Φ respects the metric between ∆ g and ∆ Φ(g) for all g ∈ G).

But the left translations respect the metric of ∆, and the one of ∆ ′ as well, so that L Φ(g) -1 • Φ • L g is an isometry from H onto H ′ . These two groups are nilpotent, and according to The Kivioja-Le Donne's Theorem again, the restriction of Φ to H is an automorphism onto H ′ .

Let Y ∈ ∆, and Y be the associated left-invariant vector field of H. Since L Φ(g

) -1 • Φ • L g is an isometry from H onto H ′ , we get that (L Φ(g) -1 • Φ • L g ) * Y is a left-invariant vector field Z of H ′ .
Let h ∈ H. Then Φ(gh) belongs to Φ(g)H ′ , and is equal to

Φ(g)h ′ for h ′ = L Φ(g) -1 •Φ•L g (h). The equality T h (L Φ(g) -1 • Φ • L g ).Y h = Z h ′ implies T gh Φ.Y gh = T gh Φ.T e L gh .Y e = T gh Φ.T h L g .Y h = T h ′ L Φ(g) .Z h ′ = T h ′ L Φ(g) T e L h ′ .Z e = T e L Φ(g)h ′ .Z e = T e L Φ(gh) .Z e .
Since g and h are arbitrary, this proves that the image by Φ * of the left-invariant vector field Y of G is a left-invariant vector field of G. This being true for all Y ∈ ∆, and according to the forthcoming lemma 3 the isometry Φ is an automorphism.

Remark. In the algebra case, the proof uses only that the groups H and H ′ are nilpotent, not that the group G itself is nilpotent.

Lemma 2 Let G be a connected Lie group and g its Lie algebra, identified with the set of leftinvariant vector fields.

Let Φ be a diffeomorphism of G that verifies Φ(e) = e and Φ * Y ∈ g for all Y ∈ g. Then Φ is an automorphism.

Proof. Let Y ∈ g be a left-invariant vector field and Z = Φ * Y . The flow of Y is (t, g) -→ g exp(tY ) and the one of Φ * Y is (t, g) -→ Φ(Φ -1 (g) exp(tY )). On the other hand Z is leftinvariant by assumption and we get:

∀g ∈ G ∀t ∈ R Φ(Φ -1 (g) exp(tY )) = g exp(tZ). (7)
Formula ( 7) applied at g = e gives Φ(exp(tY )) = exp(tZ) for all t ∈ R.

Let V be a neighborhood of e in G and U be a neighborhood of 0 in g such that exp be a diffeomorphism from U onto V . If Y ∈ U , then y = exp(Y ) ∈ V and for all g ∈ G holds:

Φ(Φ -1 (g)y) = Φ(Φ -1 (g) exp(Y )) = g exp(Z) = gΦ(exp(tY )) = gΦ(y).
In other words we have Φ(zy) = Φ(z)Φ(y) for all z = Φ -1 (g) ∈ G and all y ∈ V . Since V generates G this equality is true for all z, y ∈ G and Φ is an automorphism.

Lemma 3 Let ∆ be a subspace of g and X be a linear vector field such that the Lie algebra generated by ∆ and X be equal to g ⊕ RX .

If Φ is a diffeomorphism of G such that Φ * Y ∈ g for all Y ∈ ∆ and such that Φ * X is a linear vector field, then Φ is an automorphism.

Proof. Since the Lie bracket of a left-invariant vector field with another left-invariant vector field or a linear field is left-invariant, the Lie algebra generated by the Φ * Y , Y ∈ ∆, and Φ * X is obviously equal to g ⊕ RΦ * X . According to Lemma 2 the diffeomorphism Φ is an automorphism. Theorem 2 is no longer true in general if the group is not nilpotent. Counter-examples can be easily built with the help of the Milnor example of the rototranslation group (see [START_REF]Milnor Curvatures of left invariant metrics on Lie groups[END_REF]) recalled in [START_REF] Kivioja | Le Donne Isometries of nilpotent metric groups[END_REF].

Counter-Example. The rototranslation group is the universal covering of the group of orientationpreserving isometries of the Euclidean plane. It can be described as R 3 with the product:

  x y z   .   x ′ y ′ z ′   =   cos z -sin z 0 sin z cos z 0 0 0 1     x y z   +   x ′ y ′ z ′  
The Euclidean metric is left-invariant for this product. It can be shown that the group of automorphisms of this group that are isometries is one-dimensional though the group of isometries that preserve the identity is 3-dimensional, which implies that not all isometries are affine. Let us call this group R and let us define an ARS on G = R × R 2 in the following way: the structure of R is the previous one, and the one of R 2 is the Grushin plane. Any diffeomorphism Φ that preserves the identity, made of the direct product of an isometry of R and an isometry of the Grushin plane is an isometry of this ARS. However if the isometry of R is not an automorphism, Φ cannot be an automorphism of G.

Here G is a solvable, not nilpotent group. Many different counter-examples can be build, for instance by replacing the Grushin plane by one of the ARSs defined later on the Heisenberg group.

Conclusion

The results of this section show that an isometry that preserves the identity also preserves the left-invariant distribution, the linear field and is characterized by its tangent mapping at the identity. Moreover that last should preserve the derivation.

We are consequently interested in diffeomorphisms Φ of G that satisfy:

(i) Φ(e) = e (ii) Φ * ∆ = ∆ ′ (iii) Φ * X = X ′ (iv) T e Φ • D = D ′ • T e Φ
Since the isometries that preserve the identity are completely determined by their tangent maps at the origin we will first look for invertible linear maps P on T e G that verifies P • D = D ′ • P and P (∆ e ) = ∆ ′ e . If the Lie group G is simply connected and if such a P is an automorphism of g, it is the tangent mapping of an automorphism Φ of G. It is easy to see that this automorphism is an isometry. Indeed it transforms any left-invariant vector field into a left-invariant vector field. Since Φ satisfies P (∆ e ) = ∆ ′ e , it satisfies Φ * ∆ = ∆ ′ . On the other hand it transforms any linear vector field into a linear vector field. Since it satisfies P

• D = D ′ • P we have Φ * X = X ′ .
If either P is not an automorphism of g or the Lie group is not simply connected we cannot conclude so easily, and we have to look in each case to the existence of an isometry Φ such that T e Φ = P .

Classification.

In the two following sections we classify the ARSs on the 2D affine group and the Heisenberg group. This classification is done in three steps:

1. The ARSs are at first classified by isometry.

2. It is clear that a global rescaling of the metric does not modify the geometry. We consequently normalize the ARSs by rescaling in a second step.

3. In order to emphasize the main geometric structures we then accept to modify the leftinvariant metric in the left-invariant distribution ∆ (but we keep it Euclidean and leftinvariant). This amounts to "forget" the metric in ∆. The typically different geometries are thus exhibited.

Classification of the ARSs on the affine group

Let G be the connected 2-dimensional affine group:

G = Af f + (2) = x y 0 1 ; (x, y) ∈ R * + × R .
Its Lie algebra is solvable1 and generated by the left-invariant vector fields: gX = x 0 0 0 and gY = 0 x 0 0 where g = x y 0 1 . The linear vector field X associated to such a derivation is

In natural coordinates they write

X (g) = 0 a(x -1) + by 0 0 .
In natural coordinates it writes X (x, y) = (a(x -1) + by) ∂ ∂y . For more details, see [START_REF] Dath | Jouan Controllability of linear systems on low dimensional nilpotent and solvable Lie groups[END_REF].

An ARS on Af f + (2) is defined by a left-invariant vector field B = αX + βY and a derivation D such that B and DB are linearly independent, in order to satisfy the rank condition. In natural coordinates, the ARS is described as the system

ẋ = uαx ẏ = v(a(x -1) + by) + uβx Proposition 3 Let Σ = (X , B), ′ = (X ′ , B ′ ) be two ARSs on Af f + (2).
If Φ is an isometry between Σ and Σ and Φ(e) = e then Φ is an automorphism.

Proof. Note that Theorem 2 ensures that Φ * (∆) = ∆ ′ , Φ * (X ) = ǫX ′ , where ǫ = ±1.

Since Φ * preserves the Euclidean metric in ∆, which is 1-dimensional, we have

Φ * (B) = ǫ ′ B ′ , with ǫ ′ = ±1.
In other words Φ * Y is a left-invariant vector field for all Y ∈ ∆. Together with Φ(e) = e, and according to Lemma 3, we obtain that Φ is an automorphism.

We are now in position to state a complete classification of ARSs on Af f + (2), and to characterize them by geometric invariants.

Classification by isometries

The almost-Riemannian structures classification by isometries is simplified by two facts. First, Proposition 3 states that isometries fixing the identity are Lie group automorphism, and second Af f + (2) is simply connected. Thus, instead of searching for Lie groups automorphism, is enough to search for Lie algebra automorphisms.

Let P be an automorphism of the Lie algebra aff(2). It is easy to see that P has form

P = 1 0 c d , for c ∈ R and d ∈ R *
and is associated to the Lie group automorphism Φ(x, y) = (x, c(x -1) + dy), via the equality P = T e Φ. Proof. First note that the change of the vector fields X to -X or B to -B does not modify the metric.

The ARS is defined by B = αX + βY and D = 0 0 a b , and we have

DB = D(αX + βY ) = αDX + βDY = (αa + βb)Y.
Thus, the 2x2-matrix whose first (resp. second) column contains the coefficients of B (resp. DB) in the basis {X, Y } is given by

(B DB) = α 0 β (aα + bβ)
.

The rank condition is satisfied if and only if α(aα + bβ) = 0. Since B and DB are linearly independent, we can define an isomorphism P of the vector space g by P (B) = αX and P (DB) = αY.

This isomorphism turns out to be a Lie algebra automorphism on g. Indeed, the equality

[B, DB] = [αX + βY, (aα + bβ)Y ] = α(aα + bβ)Y = αDB, implies P [B, DB] = P (αDB) = α 2 Y = α 2 [X, Y ] = [αX, αY ] = [P B, P (DB)].
Using Theorem 2 the derivation D is transformed to the derivation P DP -1 , which is characterized by : P DP -1 X = Y and P DP -1 Y = bY.

Therefore, any ARS is isometric to Remark: The singular locus Z = Z X is a Lie subgroup of Af f + (2).

P DP -1 =
Proposition 5 The group of isometries of an ARS on Af f + (2) is the group of left translations by elements of Z X .

Proof. Let Φ be an isometry. Suppose that Φ fixes the identity. Proposition 3 ensures that any isometry Φ on Af f + (2) is an automorphism. Since Φ * (∆) = ∆, we obtain T e Φ = 1 0 0 d and using Φ * (X ) = ±X , we conclude P DP -1 = D if and only if d = 1, i.e. if and only if Φ is the identity map. Thus, the isometry group of an ARS is the group of left translations by elements of g ∈ Z X .

Global rescaling

We do not change the geometry of the ARS, if we multiply all the vector fields by a common positive constant λ. Choosing λ as 1 α , we rescale αX to X and the derivation D to

λD = 0 0 1 α b α . The ARS is isometric to 0 0 1 b α ,
for some b α ≥ 0. Therefore, any ARS is up to a rescaling isometric to an ARS defined by

B = X and D = 0 0 1 b , with b ≥ 0.
Equivalently, it is defined by B = X and the singular locus Z X = {(x, y) : (x -1) + by = 0, for some b ≥ 0}.

Moreover, the singular locus is normal if and only if b = 0, that is, Z X = {(1, y), y ∈ R}. For the other cases, that is, if b > 0 the singular locus Z X = {(x, (x-1) b ), x ∈ R} is not normal. Moreover, the singular locus Z is a normal Lie subgroup if and only if b = 0 and not normal otherwise.

Conclusion

There two main models, the ones obtained above after rescaling and modification of the leftinvariant metric, are completely characterized by the singular locus.

The case where the singular locus is a normal subgroup, that is the the case b = 0, has been completely studied in [START_REF] Ayala | Jouan Almost-Riemannian Geometry on Lie groups[END_REF].

The other one remains to be analyzed.

Classification of the ARSs on the Heisenberg group

Let G be the 3-dimensional Heisenberg group:

G =      1 x z 0 1 y 0 0 1   ; (x, y, z) ∈ R 3    .
Its Lie algebra g is nilpotent2 and generated by the left-invariant vector fields:

gX =   0 x 0 0 0 0 0 0 0   , gY =   0 0 x 0 0 1 0 0 0   and gZ =   0 0 z 0 0 0 0 0 0   where g =   1 x z 0 1 y 0 0 1   .
They verify [X, Y ] = Z and the other brackets vanish. In the basis (X, Y, Z), all derivations D of g have the form

D =   a b 0 c d 0 e f a + d  
, where a, b, c, d, e, f ∈ R.

The linear vector field X associated to such a derivation is

X (x, y, z) = (ax + by) ∂ ∂x + (cx + dy) ∂ ∂y + (ex + f y + (a + d)z + 1 2 cx 2 + 1 2 by 2 ) ∂ ∂z .
For more details, see [START_REF] Dath | Jouan Controllability of linear systems on low dimensional nilpotent and solvable Lie groups[END_REF]. An ARS on G is defined by an orthonormal frame {B 1 , B 2 , X }, where B 1 , B 2 are left-invariant vector fields and X is a linear one with associated derivation D.

To classify the ARSs by isometries, note that Theorem 4 states that all isometries that fix the identity are Lie group automorphisms. Since the Heisenberg group is simply connected, is enough to work with Lie algebra automorphisms.

As we will see, there are two very different cases according to whether ∆ is a subalgebra or not.

∆ is a subalgebra

Classification by isometries

An automorphism of the Lie algebra g, in the basis (X, Y, Z), has the form:

P =   α β 0 γ δ 0 ǫ ζ det(A)   , for A equal to α β γ δ , and
det(A) = 0. Note that Z belongs to ∆. Indeed, suppose B 1 = a 1 X +b 1 Y +c 1 Z and B 2 = a 2 X +b 2 Y +c 2 Z, the first two coordinates are linearly dependent, because ∆ is a subalgebra. Thus there exists λ ∈ R such that (a 1 , b 1 ) = λ(a 2 , b 2 ). If λ = 0 than B 2 = c 2 Z, otherwise take the linear combination X -λY = (c 1 -λc 2 )Z.
So, we can choose the orthonormal frame of ∆ as {B 1 , ηZ}, for some η ∈ R * .

Proposition 6 Any almost-Riemannian structure on G, whose distribution ∆ a subalgebra, is isometric to an almost-Riemannian structure whose orthonormal frame is {X, Z, X } and the derivation D has the following form:

D =   0 b 0 c d 0 0 f d   ,
for some c > 0 and d, f ≥ 0. Moreover, two different ARSs of this form are not isometric.

Proof. Let P be the isomorphism defined by P (B 1 ) = X, P (DB 1 ) = a 1 η Y and P (ηZ) = Z. The isomorphism P is an automorphism because

P [B 1 , Z] = 0 = [X, ηZ] = [P (B 1 ), P (Z)]. P [B 1 , Y ] = P (a 1 Z) = a 1 η Z = [X, a 1 η Y ] = [P (B 1 ), P (DB 1 )].
The derivation D = P DP -1 associated to X verifies DX = P DP -1 X = P DB = a 1 η Y. Hence, in relation to the basis (X, Y, Z) it has the form

D =   0 b 0 c d 0 0 f d   where c = a 1 η = 0.
The changes X to -X and B to -B allow to assume d ≥ 0. Take

P ǫǫ ′ =   ǫ 0 0 0 ǫ -1 ǫ ′ 0 0 0 ǫ ′   , where ǫ, ǫ ′ = ±1.
We can apply P ǫǫ ′ to change b to ǫǫ ′ b, c to ǫǫ ′ c and f to ǫf. Thus, we can assume c > 0 and d, f ≥ 0. It is easy to see that two ARSs related to different derivations of this form are not isometric.

Proposition 7

The group of isometries of an ARS in the form of Proposition 6 is: If d = 0 f = 0 the ARS group of isometries is composed only of left translations by elements of Z X .

If d = 0 f = 0 the ARS group of isometries is composed of left translations by elements of Z X and infinitesimal isometries as P ǫǫ ′ for ǫ = ±1 and ǫ ′ = 1.

If d = 0 f = 0 the ARS group of isometries is composed of left translations by elements of Z X and infinitesimal isometries as P ǫǫ ′ for ǫ = ǫ ′ = ±1.

If d = f = 0 the ARS group of isometries is composed of left translations by elements of Z X and infinitesimal isometries as P ǫǫ ′ for ǫ, ǫ ′ = ±1.

Generically b, d and f are nonzero and the set Z X is reduced to the identity. Thus in the subalgebra case, the group of isometries is generically reduced to the identity.

Proof. Let Φ be an isometry fixing the identity. It is an automorphism. Let T e Φ = P ǫǫ ′ . Note that P ǫǫ ′ (X) = ǫX and P ǫǫ ′ (Z) = ǫ ′ Z. We obtain that P ǫǫ ′ changes D to

P ǫǫ ′ DP -1 ǫǫ ′ =   0 ǫ ′ b 0 ǫ ′ c d 0 0 ǫf d   . (8)
Since Φ * (X ) = ±X we have P ǫǫ ′ D(P ǫǫ ′ ) -1 = ±D. Generically b, c, d and f are non zero, so ǫ and ǫ ′ need to be equal to 1. Moreover a straightforward computation proves that Z X is reduced to {e}. As a consequence the group of isometries is generically reduced to the identity.

Considering Equation [START_REF] Bonnard | The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry[END_REF] it is easy to conclude in the other cases.

Global rescaling

We do not change the geometry of the ARS if we multiply all the vector fields by a common positive constant λ. This global rescaling allows us to use

P =   α 0 0 0 ǫ 0 0 0 αǫ   for α ∈ R * and ǫ = ±1, to changes D into D =   0 αǫ -1 b 0 α -1 ǫc d 0 0 αf d   .
Therefore, any ARS is isometric up to a rescale to one and only one ARS defined by the orthonormal frame {X, Z, X } where the associated derivation D is equal to:

D =   0 b 0 1 d 0 0 f d   5.1.

Deformation of the Euclidean metric in ∆

In order to obtain a classification with less parameters, we consider as equivalent two ARSs that have the same linear fields, same distributions but provided with a different left-invariant metrics.

In other words, we forget the metrics. Let Σ be the ARS defined by the orthonormal frame {X, Z, X }, where the associated derivation is

D =   0 b 0 1 d 0 0 f d   .
We consider as equivalent to Σ, any ARS obtained by rescaling and conjugation by automorphisms that preserve ∆, that is:

P =   α β 0 0 δ 0 ǫ ζ αδ   .
Thus, we can conclude that, any almost-Riemannian structure on G, whose left-invariant distribution ∆ is a Lie subalgebra, is related to one and only one ARS whose orthonormal frame is {X, Z, X }, and as previously the associated derivation D has the form:

D =   0 b 0 1 d 0 0 f d   ,
but the constants b, d and f are normalized as follows:

case d b f singular locus Z Z X eigenvalues l 1 , l 2 (i) 1 > -1 4 0 x+y=0 (0,0,0) l 1 = 1+ √ 1+4b 2 l 2 = 1- √ 1+4b 2 (ii) 1 < -1 4 0 x+y=0 (0,0,0) l 1 = 1+ √ -(1+4b)i 2 l 2 = 1- √ -(1+4b)i 2 (iii) 1 0 1 x+y=0 (-y, y, -y -1 2 y 2 ) l 1 = 0 l 2 = 1 (iv) 1 0 0 x+y=0 (-y, y, -1 2 y 2 ) l 1 = 0 l 2 = 1 (v) 0 1 0 x=0 x=y=0 l 1 = 1 l 2 = -1 (vi) 0 -1 0 x=0 x=y=0 l 1 = i l 2 = -i (vii) 0 0 1 x=0 x=y=0 l 1 = l 2 = 0 (viii) 0 0 0 x=0 x=0 l 1 = l 2 = 0
Remark: For this case, we know that Z is always a Lie subgroup and its Lie algebra T e Z = D -1 ∆. The Lie subalgebra ∆ is not tangent to the singular locus. Therefore, there are no tangency points when ∆ is a subalgebra.

∆ is not a subalgebra

In this case the distribution ∆ generates the Lie algebra. So, to satisfy the rank condition there are no restriction on the derivation D. On the other hand, Condition (iii) of Definition 1 states that X and ∆ need be linearly independent in an open and dense set.

Classification by isometries

Proposition 8 Any ARS, whose distribution ∆ is not a subalgebra, is isometric to an ARS whose orthonormal frame is {X, Y, X }, where the derivation D has the following form:

D =   a b 0 c d 0 0 f a + d   with c, f ≥ 0.
The conditon that X (g) does not belong to ∆(g) everywhere reduces to:

If b = c = f = 0 then a + d = 0. Proof. If ∆ is not a subalgebra then [B 1 , B 2 ] = (a 1 b 2 -a 2 b 1 )Z / ∈ ∆.
Let us choose the isomorphism P defined by

P (B 1 ) = X, P (B 2 ) = Y and P Z = 1 (a 1 b 2 -a 2 b 1 ) Z.
Actually, the isomorphism P is a Lie algebra automorphism

[P (B 1 ), P (B 2 )] = [X, Y ] = Z = P ((a 1 b 2 -a 2 b 1 )Z) = P [B 1 , B 2 ].
To finish, we can apply a rotation within ∆ in order to simplify the derivation D. Consider

P θ,ǫ = R θǫ 0 0 det(R θǫ ) =   cos(θ) -ǫ sin(θ) 0 sin(θ) ǫ cos(θ) 0 0 0 ǫ   ,
where R θǫ is an orthogonal transformation of ∆ and P ǫ,ǫ ′ is defined in previous section. Case (i) e, f = 0, c ∈ R. Apply P θ,1 for some θ to vanish e then apply P ǫ,ǫ ′ to transform f to ǫf and c to ǫ ′ c. So, we can assume c, f ≥ 0.

Case (ii) e = 0, f = 0 and c ∈ R. Apply P ǫ,ǫ ′ to transform f to ǫf and c to ǫ ′ c. So, we can assume c, f ≥ 0.

Case (iii) e, f = 0 and c ∈ R. Apply P ǫ,ǫ ′ to transform c to ǫ ′ c so we can assume c ≥ 0.

Proposition 9

The group of isometries of such an ARS is generated by the left translations by elements of Z X and some of the automorphisms P θ,ǫ defined above. Generically, the group of isometries is reduced to left translations.

Proof. Let Φ be an isometry fixing the identity. Theorem 4 ensures that Φ is an automorphism. Since Φ preserves the Euclidean metric in ∆, we have Φ * (X) = αX + ǫβY, Φ * (Y ) = -βX + ǫαY and Φ * (X ) = ǫ ′ X , with α 2 + β 2 = 1, and ǫ, ǫ ′ = ±1. Note that the differential Φ * at the identity has the form of the Lie algebra automorphism P θ,ǫ . The derivation

D = A 0 (e, f ) tr(A) is changed by P θǫ to P θǫ DP -1 θǫ = R θǫ AR -1 θǫ 0 ǫ(e, f )R -1 θǫ tr(A)
.

Note that for e, f = 0, P θǫ ′ alters (e, f ), except when θ = 2kπ for k ∈ Z and the automorphism P ǫǫ ′ change b to ǫ ′ b, c to ǫ ′ c, e to ǫǫ ′ c and f to ǫf. If (e, f ) = (0, 0) to preserve D we need ǫ = ǫ ′ = 1. Generically tr(A), e, f = 0 and the only solution is P θǫ ′ = I.

Global rescaling

We do not change the geometry of the ARS, if we multiply all the vector fields by a common positive constant λ. This global rescaling allows us to use

P =   α 0 0 0 α 0 0 0 α 2   where α ∈ R * , to changes D into D =   a b 0 c d 0 0 α -1 f a + d   .
Therefore, any ARS is defined up to a rescale by the orthonormal frame {X, Z, X }, where the derivation D associated to X has one of the following form: If f = 0 we change it to 1. If f = 0 and c = 0 we change it to 1. If c, f = 0 and b = 0 we change it to ±1. If b, c, f = 0 and a + d is different from 0, so we change it to 1.

Deformation of the Euclidean metric in ∆

In order to get a classification with less parameters, we consider as equivalent ARS that have the same linear fields, same distributions but provided with a different left-invariant metrics.

In other words, we forget the left-invariant metric on ∆. Consider the orthonormal frame of the previous section.

We consider as equivalent to Σ any ARS obtained by rescaling and conjugation by automorphisms that preserve ∆, that is:

P =   α β 0 γ δ 0 0 0 αδ -βγ   .
The Lie algebra automorphism P transforms the ARS (Σ) to the ARS ( Σ) defined by the orthonormal frame {X, Z, X } where X is associated to the derivation A short computation proves the impossibility to vanish e and f simultaneously. We use the diagonalization process to classify by eigenvalues. By doing so, we may alter the entries e and f again. Thus, first we need to put the derivation D in a block form and then apply the previous computations to obtain the following forms: ef 2l 1 ). In case (2.ii, the function g is g(x, y, z) = ex + f y + 1 2

y 2 . ( 13
)
The differential dg is surjective at all points only if e is different from zero. Under this assumption the singular locus Z is a connected two dimensional manifold. If e is zero, there are many points where the application dg is not surjective. In this case, the set g -1 (0) has different behaviours according to the value of f. If f = 0, the set g -1 (0) is the union two planes perpendicular to the y -axis, one through the point (0, 0, 0) and the other through the point (0, -2f, 0). If f = 0, the set g -1 (0) is a plane perpendicular to the y-axis passing through the point (0, 0, 0).

Thus, if e = 0, there are no tangency points. If e = 0, the tangency points need to satisfy the equation f + y = 0. So, for f = 0, there are no tangency points. If f = 0 all the singular locus is equal to the set of tangency points.

In case (3.i) the function g is g(x, y, z) = ex + f y + 2az + 1 2 bx 2 + 1 2 by 2 -axy.

The differential dg is surjective at all points of the singular locus. So, Z is a connected two dimensional manifold. Since the following linear system solution is unique e + bx -ay = 0 and f + by + ax = 0, there is one tangency point.

In case (3.ii), the function g is g(x, y, z)

= ex + f y + 1 2 bx 2 + 1 2 by 2 . ( 15
)
The differential dg is not surjective, but the equality ex + f y + 1 2 bx 2 + 1 2 by 2 = 0 represents an elliptic cylinder when e or f is different from 0. If e = f = 0, it is a line through the identity. If e or f is different from 0, there are no tangency points. If e and f both vanish, the tangency points forms the line (0, 0, z) z ∈ R.

In order to simplify even more the derivation, is possible to apply one or both of the following two kinds of automorphisms:

  X(x, y) = x ∂ ∂x and Y (x, y) = x ∂ ∂y . They verify [X, Y ] = XY -Y X = Y . In the basis (X, Y ), all derivations D of the Lie algebra aff(2) have the form D = 0 0 a b , where a, b ∈ R.

Proposition 4

 4 Any almost-Riemannian structure on the Lie group Af f + (2) is isometric to one and only one of the structures defined by D = 0 0 1 b and B = αX, where α > 0 and b ≥ 0.

  αX with α > 0 and b ≥ 0. It is easy to check that two different such ARSs are not isometric.

4. 3 0 1 1 .

 31 Deformation of the Euclidean metric in ∆ Deforming the left-invariant metric, we can assume B = bX, (if b > 0) and the derivation D = 0 0 b b , which are isometric up to a rescale to B = bX and 0 Therefore, any almost-Riemannian structure on the Lie group Af f + (2) is related by B = X and the derivation

  + βc) -γ(αb + βd) -β(αa + βc) + α(αb + βd) 0 δ(γa + δc) -γ(γb + δd) -β(γa + δc) + α(γb + δd) 0 δe -γf -βe + αf (αδ -βγ)(a + d)   .

l 1 cx 2 + 1 2 by 2 - 2 y 2 --e l 2 1 , e l 1 , -e 2 4l 2 1

 1222111 , l 2 , a, b, e, f are real numbers, b = 0 and a 2 + b 2 > 0.For ∆ = {X, Y } and D = the singular locus is described by the followingformula Z = {ex + f y + (a + d)z -1 2 dxy = 0}. In case (2.i) the function g is g(x, y, z) = ex + f y + 2l 1 z + 1 l 1 xy,and 0 is a regular value. So, the singular locus is a connected two dimensional manifold. This ARS has one tangency point given by (x, y, z) = (-f l 1

  case, together with a global rescaling. We obtain that any almost-Riemannian structure on G, whose left-invariant distribution ∆ is not a Lie subalgebra, is related to only one ARS whose the orthonormal frame is {X, Y, X } where the derivation D has the following form:

A Lie algebra g is solvable if its derived series terminates in the zero Lie algebra: define D 1 g = [g, g] and by induction D n+1 g = [D n g, D n g], then g is solvable if D n g vanishes for some integer n.

A Lie algebra g is nilpotent if its central series terminates in the zero Lie algebra: define C 1 g = [g, g] and by induction C n+1 g = [C n g, g], then g is nilpotent if C n g vanishes for some integer n.

The case [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian Geometry[END_REF] have four distinct subcases with distinct characteristics: case l 1 l 2 l 1 + l 2 singular locus Z Z X (i) = 0 = 0 = 0 (l 1 + l 2 )z = l 2 xy -ex -f y x = y = z = 0 (ii) = 0 -l 1 0 0 = l 1 xy + ex + f y x = y = 0 (iii) = 0 0 l 1 l 1 z = -ex -f y (0, y, f l 1 y) (iv) 0 0 0 0 = ex + f y 0 = ex + f y

The case [START_REF] Agrachev | A Gauss-Bonnet like formula on two-dimensional almost-Riemannian manifolds[END_REF] has two different behaviours: cases

The case (3) has two different behaviours and a restriction a 2 + b 2 > 0:

Tangency points. It is clear that an isometry sends tangency points to tangency points and that to find them it is necessary to describe the singular locus.

Notice in case (1.i), the singular locus Z is g -1 (0), where the function g is

The differential dg is surjective at all points of the locus that is, 0 is a regular value of the function g. So, the inverse image g -1 (0) is a connected two dimensional manifold. This case has one tangency point

The differential dg is not surjective at all points of the singular locus when f or e is equal to zero.

If e, f = 0, the singular locus Z is a nonconnected two dimensional manifold. If f = 0, the singular locus is the union two planes intersecting in the line (0, e l 1 , z). If e = 0, the singular locus is the union two planes intersecting in the line ( f l 1 , 0, z). If e, f = 0 the orthonormal frame {X, Y, X } does not describe an ARS.

Tangency points. This case has three different behaviours. If e and f are both nonzero, there are no tangency points, if e = 0 tangency points form a line ( f l 1 , 0, z), if f = 0 tangency points form a line (0, e l 1 , z). In these cases the respective lines are the intersection of the two planes. In case (1.iii) the function g is

Note that the function g describes a plane. So, the singular locus is a connected two dimensional manifold.

Tangency points. This case has two possible behaviours. If e is zero, the tangency points form a 1-dimensional manifold described as ( -f l 1 , y, -f y l 1 ), y ∈ R. If e is not zero, there are no tangency points.

In case (1.iv) the function g is

Note that the function g describes a plane. So, the singular locus is a connected two dimensional manifold. If e and f are both zero then D is the zero derivation, which do not describe an ARS. There are no tangency points. points of G -Z 3.i.1 1 R * (0, 1) submanifold one tangency point two 3.i.2 1 R * (0, 0) submanifold (0,0,0) two 3.ii.1 0 1 (0, 1) submanifold no tangency points two 3.ii.2 0 1 (0, 0) Lie subgroup the line x = y = 0. one