Varun Jain
email: varun.jain@inria.fr

Evanthia Mavridou
email: evanthia.mavridou@inria.fr

James L Crowley
email: james.crowley@inria.fr

Augustin Lux
email: augustin.lux@inria.fr

Facial Expression Analysis and the Affect Space

Keywords: Automated Facial Expression Analysis, Affect Recognition, Multi-scale Gaussian Derivatives

In this paper we present a technique for facial expression analysis and representing the underlying emotions in the affect space. We develop a purely appearance based approach using Multi-scale Gaussian derivatives and Support Vector Machines. The technique is validated on two different databases. The system is shown to generalize well and performs better than the baseline method.

I. INTRODUCTION

Facial expressions are a mirror to human emotions and an important component of human to human interaction. Human computer interaction requires the same ability to read emotions from facial expressions. Ekman introduced the concept of six basic emotions that are universally recognizable [START_REF] Ekman | An argument for basic emotions[END_REF]. In [START_REF] Ekman | Facial Action Coding System: A Technique for the Measurement of Facial Movement[END_REF] he presented the Facial Action Coding System(FACS), a taxonomy to describe facial expressions in terms of individual muscle movements. FACS based approaches have been adopted in a variety of vision systems such as the Computer Expression Recognition Toolbox(CERT) [START_REF] Littlewort | The computer expression recognition toolbox (cert)[END_REF]. Such systems are trained to estimate the Action Unit(AU) intensities which can then be used to assign one of the six basic emotion labels to that image or frame. The problem arises when the expression in the image is not associated with any of the six basic emotions. An alternative to such a structured approach is to represent the underlying emotions in a multidimensional emotion space. Shin in [START_REF] Shin | Recognizing facial expressions with pca and ica onto dimension of the emotion[END_REF] uses component analysis techniques to recognize emotions and map them to the affect space. Another method was presented by Dahmane and Meunier in [START_REF] Dahmane | The computer expression recognition toolbox (cert)[END_REF]. The authors used Gabor wavelets and Support Vector Machines on the Semaine database [START_REF] Mckeown | The semaine database: Annotated multimodal records of emotionally colored conversations between a person and a limited agent[END_REF] and they use 4 dimensions (Activation, Expectation, Power and Valence) to represent the emotions that underlie the facial expressions. In [START_REF] Russell | Evidence for a three-factor theory of emotions[END_REF] the authors argue that three dimensions are enough to represent any emotion. In this paper we use the affect space model developed by Russell and Mehrabian and compare our results for Pleasure and Arousal with the results from the technique presented in [START_REF] Dahmane | The computer expression recognition toolbox (cert)[END_REF]. Two common ways to describe image features are: appearance based methods and geometric feature based methods. The latter involves detection and tracking of facial keypoints such as the lip corners, nostrils and eyes. This detection and tracking is done with the help of computationally expensive vision techniques and are not very robust. The approach we present here does not involve identification of any landmarks on the face and just like the appearance based technique discussed in [START_REF] Dahmane | The computer expression recognition toolbox (cert)[END_REF], the image filters are applied to the whole-face to obtain the feature vector.

II. AFFECT SPACE AND DATASETS USED

Russell and Mehrabian in [START_REF] Russell | Evidence for a three-factor theory of emotions[END_REF] describe a 3-dimensional affect space(Pleasure/Displeasure, Arousal/Sleep, Dominance/Submissiveness) model that can be used to describe the emotional state of a person. Experiments support that these 3 dimensions are sufficient to represent all human emotions. The Pleasure/Displeasure Scale measures the pleasantness of an emotion while the Arousal/Sleep Scale measures the intensity of the emotion and the Dominance/Submissiveness Scale represents the controlling and dominant nature of the emotion. The third axis of Dominance remains controversial and there is evidence to suggest that there is a high correlation between dominance and the other two axes [START_REF] Becker-Asano | WASABI:Affect Simulation for Agents with Believable Interactivity[END_REF]. Our approach was tested on the Cohn-Kanade [START_REF] Cohn | Comprehensive database for facial expression analysis[END_REF] and FEED [START_REF] Wallhoff | Facial expressions and emotion database[END_REF] datasets. The FEED dataset was collected at the Technical University of Munich. The dataset was generated as a part of the European Union FG-NET project [11].

The FEED dataset does not contain posed emotions, the emotions were elicited by showing video clips to the participants. The database contains images from 18 individuals for 6 basic emotions along with the neutral face.

III. MULTI-SCALE GAUSSIAN DERIVATIVES

Gaussian derivatives can efficiently describe the neighborhood appearance of a pixel for pattern recognition tasks [START_REF] Jain | Smile detection using multi-scale gaussian derivatives[END_REF]. For images this is done by calculating different orders of Gaussian derivatives normalized in scale and orientation at every pixel. The basic Gaussian function is defined as:

G(x, y; ) = e x 2 +y 2 2 2
(1)

Here is the scale factor or variance and defines the spatial support. This function measures the intensity of the neighborhood and does not contribute to the identification of the neighborhood and can be omitted. First order derivatives provide information about the gradient (intensity and direction) whereas the second order derivatives provide the information about image features such as bars, blobs and corners. Higher order derivatives are only useful if the second order derivatives are strong otherwise they just contain image noise.

Obtaining scale invariant features is not a trivial task. Several methods have come up in the past addressing this problem. Lindeberg in [START_REF] Lindeberg | Scale-Space Theory in Computer Vision[END_REF] suggests that Gaussian derivatives be calculated across scales to get scale invariant features and then Lowe in [START_REF] David | Object recognition from local scale-invariant features[END_REF] defines the intrinsic or characteristic scale as the value of the scale parameter at which the Laplacian provides a local maximum. The computational cost of directly searching the scale axis for this characteristic scale can be prohibitively expensive. A cost-effective method for computing Multi-scale Gaussian derivatives has been discussed in detail in [START_REF] Crowley | chapter Fast Computation of Scale Normalized Gaussian Receptive Fields[END_REF]. The next section is about Principal Component Analysis (PCA) and why we need it.

IV. PRINCIPAL COMPONENT ANALYSIS

The region of the image containing the face is normalized to 64 X 64 pixels, this particular size is chosen after extensive experimentation where normalized images of 64 X 64 pixels gave the best accuracy. We calculate several orders of derivatives at 2 levels of scale for every pixel but it leads to an enormous feature vector. Therefore we divide the image into cells of 4 X 4 pixels and the feature vector contains the mean and standard deviation of the descriptor values (gaussian derivatives) for each cell of 4 X 4 pixels. Principal Component Analysis is used for dimensionality reduction which reduces the prediction time when the Support Vector Machines are used for classification. Correlation in the data can be reduced by transforming the original dimensions into new dimensions which are a linear sum of the original dimensions but are linearly uncorrelated. Then these new dimensions are ranked according to the variance i.e. the dimension which accounts for the most variability in the data gets the first rank and so on [START_REF] Jolliffe | Principal Component Analysis[END_REF]. PCA is done by eigenvalue decomposition of the data correlation matrix after normalizing the data for each dimension. PCA provides us with scores and loadings. The scores are the transformed values corresponding to the data point and loadings are the coefficients the original variable should be multiplied with to get the score.

V. SUPPORT VECTOR MACHINES

Support Vector Machines (SVM) belong to a family of nonprobabilistic linear classifiers [START_REF] Vapnik | Statistical Learning Theory[END_REF]. The Radial Basis kernel provides the best accuracy for the particular application and is represented the following equation:

K(x i , x j ) = e ||x i x j || 2 2 2 (2)
We use a soft margin SVM. Soft margin SVMs are used when the classes are not separable even after transforming the data to a higher dimension. The condition for the optimal hyper-plane can be relaxed by including an extra term ⇠ [START_REF] Cortes | Machine Learning, volume 20, chapter Support-Vector Networks[END_REF]:

y i (X T i W + b) 1 ⇠ i , (i = 1, ..., m) (3)
For minimum error,⇠ i should be minimized as well as ||W ||, and the objective function becomes:

minimize W T W + C m X i=1 ⇠ k i subject to y i (X T i W + b) 1 ⇠ i , and ⇠ i 0; (i = 1, ..., m) (4)
Here C is a regularization parameter that controls the trade-off between maximizing the margin and minimizing the training error. 1/ or is the width of the radial basis kernel. The C-penalty parameter is chosen using cross validation. For the data in hand C = 15 and = 280 lead to the highest classification accuracy for Arousal and C = 10 and = 190 are found to be the optimum parameters for Pleasure.

VI. THE APPROACH

Face detection is performed on the images in the dataset using the OpenCV face detector [START_REF] Viola | Robust real-time face detection[END_REF]. Following that a halfoctave gaussian pyramid is constructed over a normalized imagette of the face. This is followed by dimensionality reduction by PCA and regression using Support Vector Machines. The figure below illustrates the process. The accuracy of our approach over the Cohn-Kanade set is 85.32,82.06 percent for pleasure and arousal respectively. On the other hand the approach developed by Dahmane and Meunier achieves an accuracy of only 71.80,74.94 percent for pleasure and arousal respectively. We also see that it takes much less time to computer Gaussian derivatives using the half-octave pyramid as compared to Gabor features because of the ineteger coefficient Half-Octave Pyramid used. The table below shows the time to calculate the features for the complete Cohn-Kanade database using the two techniques on the same machine(Intel Xeon Quad-Core 3GHz, 4GB RAM). Our approach is then tested on the FEED database and the accuracy for Pleasure-Displeasure is 70.73% while it is 70.08% for Arousal-Nonarousal.

Fig. 1 :

 1 Fig. 1: Example Images from the FEED dataset

Fig. 2 :

 2 Fig. 2: The image divided into cells of 4 X 4 pixels.

Fig. 3 :

 3 Fig. 3: Graph of Classification Accuracy vs. C-parameter and for Pleasure.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Graph of Classification Accuracy vs. C-parameter at = 81 for Pleasure.

Fig. 6 :

 6 Fig. 6: Graph of Classification Accuracy vs. C-parameter at = 81 for Arousal.

Fig. 7 :

 7 Fig. 7: Schematic of our approach.

Fig. 8 :

 8 Fig. 8: ROC of the classifier for Pleasure

Fig. 10 :

 10 Fig. 10: Comparison of results

Fig. 11 :

 11 Fig. 11: Results on the FEED database

TABLE II :

 II Comparison of time required for calculating the two types of featuresPCA reduces the prediction time by a factor of over 60, table3compares the prediction time with and without using PCA.

		SVM	SVM
		with PCA without PCA
	Prediction time(sec)	0.0155	0.8495

TABLE III :

 III Comparison of prediction time with and without using PCA Table4shows the prediction time of our technique versus the state of the art because our feature vector is much smaller.

		Our	State of
		Approach	the art
	Prediction Time(sec)	0.0155	1.06

TABLE IV

 IV

: Comparison of prediction time

We have presented a novel method to analyze facial expressions and represent the underlying emotion in the affect space. Not only is our performance better than that of the baseline approach, it is also faster at descriptor calculation and prediction. The approach performs better than the benchmark technique and is easily adaptable to mobile systems. Codes exist for calculating Multi-scale Gaussian derivatives on embedded systems using only integer coefficients.


Georges Dloussky

BIHERMITIAN METRICS ON HOPF SURFACES

. We also show that the Inoue complex surfaces with b2 = 0 do not admit bihermitian metrics. This completes the classification of the compact complex surfaces admitting strongly bihermitian metrics.

This paper was written while the first named author was visiting the University of Aix-Marseille 1.

Introduction

A bihermitian structure on a 4-dimensional connected manifold M consists of a pair of integrable complex structures J + and J -inducing the same orientation, and a Riemannian metric g which is hermitian with respect to both J ± . As a trivial example one can take a genuine hermitian structure (g, J) and put J ± = ±J. We exclude this situation by assuming that J + (x) = ±J -(x) at at least one point x of M . The special case when J + = J -everywhere on M will be referred to as strongly bihermitian structure.

In pure mathematics literature the theory of bihermitian 4-manifolds was initiated [START_REF] Salamon | Special structures on 4-manifolds[END_REF][START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF][START_REF] Kobak | Explicit doubly-Hermitian metrics[END_REF][START_REF] Apostolov | The Riemannian Goldberg-Sachs Theorem[END_REF] from the point of view of 4-dimensional conformal geometry, where the two integrable almost-complex structures are given by the roots of the conformal Weyl tensor. Subsequent work [START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF][START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF][START_REF] Dloussky | On surfaces of class VII + 0 with numerically anticanonical divisor[END_REF] was mainly focussed on answering the following Question 1. When does a compact complex surface (M, J) admit a bihermitian structure (g, J + , J -) with J = J + ? There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kähler geometry, introduced and studied by Gualtieri [START_REF] Gualtieri | Generalized complex geometry[END_REF] in the context of the theory of generalized geometric structures initiated by Hitchin [START_REF] Hitchin | Generalized Calabi-Yau manifolds[END_REF]. It turns out [START_REF] Gualtieri | Generalized complex geometry[END_REF] that a generalized Kähler structure is equivalent to the data of a Riemannian metric g and two g-orthogonal complex structures (J + , J -), satisfying the relations

d c + F g + + d c -F g -= 0, dd c ± F g ± = 0, where F g ± (•, •) = g(J ± •, •
) are the fundamental 2-forms of the hermitian structures (g, J ± ), and d c ± are the associated i(∂ ± -∂ ± ) operators. (These conditions on a pair of hermitian structures were, in fact, first described in the physics paper [START_REF] Gates | Twisted multiplets and new supersymmetric nonlinear sigma models[END_REF] as the general target space geometry for a (2, 2) supersymmetric sigma model.) In four dimensions we obtain a bihermitian structure, provided that the generalized Kähler structure is of even type (which corresponds to our assumption that J ± induce the same orientation on M ) and F g + = ±F g -(i.e. J + = ±J -).

When the first Betti number of M is even, it follows from [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF] (see also Remark 2 below) that, conversely, any bihermitian metric arises from a generalized Kähler structure, up to a conformal change of the metric. This correspondence was used to produce a number of new constructions of bihermitian metrics [START_REF] Goto | Deformations of generalized complex and generalized Kähler structures[END_REF][START_REF] Bursztyn | Reduction of Courant algebroids and generalized complex structures[END_REF][START_REF] Hitchin | Instantons and generalized Kähler geometry[END_REF][START_REF] Lin | Symmetries in generalized Kähler geometry[END_REF], where the existence of a Kähler structure on M plays a crucial rôle.

When the first Betti number b 1 (M ) is odd, on the other hand, the manifold (M, J) does not admit Kähler metrics, nor a bihermitian conformal structure necessarily come from a generalized Kähler structure. (As a matter of fact no strongly bihermitian metric can be obtained from a generalized Kähler structure, see Remark 2). Thus, examples of bihermitian structures in this case are still scarce (see, however, [START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]).

Thus motivated, we prove in this note the following result. Theorem 1. A compact complex surface (M, J) with odd first Betti number admits a strongly bihermitian structure (g, J + , J -) with J + = J if and only if it is a Hopf surface whose canonical bundle defines a class in the image of

H 1 (M, R * + ) → H 1 (M, O * ).
These are the Hopf surfaces with universal covering space C 2 \ {(0, 0)} and fundamental group Γ which belongs to one of the following cases:

(a) Γ is generated by the automorphism (z 1 , z 2 ) → (αz 1 , ᾱz 2 ) (where α ∈ C, 0 < |α| < 1), and a finite subgroup of SU (2); (b) Γ is generated by the automorphisms (z 1 , z 2 ) → (αz 1 , aα -1 z 2 ) (where α ∈ C, 0 < |α| 2 < a < |α| < 1), and (z 1 , z 2 ) → (εz 1 , ε -1 z 2 ) with ε primitive ℓ-th root of 1; (c) Γ is generated by the automorphisms (z 1 , z 2 ) → (β m z 1 + λz m 2 , βz 2 ) (where λ ∈ C * , β m+1 = a, 0 < a < 1), and (z 1 , z 2 ) → (εz 1 , ε -1 z 2 ) with ε primitive ℓ-th root of 1, where m = kℓ -1, k ∈ N * .
Note that the Hopf surfaces with fundamental group belonging to the case (a) of the above theorem are precisely those which admit a compatible hyperhermitian structure (see [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF]). A description of all possible finite subgroups of SU (2) that appear in this case can be found in [START_REF] Kato | Compact Differentiable 4-folds with Quaternionic structures[END_REF].

Combined with the results in [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF], Theorem 1 yields the complete list of the compact complex surfaces (M, J) which carry a strongly bihermitian structure with J + = J: these are K3 surfaces, complex tori, and the Hopf complex surfaces of the type described above. Note that in the case when b 1 (M ) is even, (M, J) does also admit a compatible hyperhermitian structure [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF], but this is not longer true when b 1 (M ) is odd.

Compared to the partial results in [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF], Theorem 1 brings in two new ingredients. First of all, we show that the Inoue surfaces with b 2 (M ) = 0 (defined and studied in [START_REF] Inoue | On surfaces of class VII0[END_REF]) do not admit compatible bihermitian structures. This is achieved by combining a recent result of A. Teleman [START_REF] Teleman | The pseudo-effective cone of a non-Kählerian surface and applications[END_REF]Rem. 4.2] (according to which on any Inoue surface the degree of the anti-canonical bundle is negative with respect to any standard hermitian metric) together with an observation from [1, Rem. 1] (that this degree must be positive, should a bihermitian structure exists).

Secondly, we address the question of existence of bihermitian structures on the Hopf surfaces in our list. We give a concrete construction of bihermitian metrics by using an idea of Hitchin [START_REF] Hitchin | Bihermitian metrics on Del Pezzo Surfaces[END_REF]. It allows us to deform the locally conformally Kähler metrics (g GO , J) found by Gauduchon-Ornea [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF] in order to obtain a family of bihermitian metrics (g t , J, J t -) with J t -= φ * t (J), where φ t is a path of diffeomorphisms; as t → 0, J t -→ J and g t /t → g GO .

Bihermitian geometry -preliminary results

In this section we recall some properties of bihermitian metrics, which we will need for the proof of our main results. The articles [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF][START_REF] Hitchin | Bihermitian metrics on Del Pezzo Surfaces[END_REF] are relevant references for more details.

Let (g, J + , J -) be a bihermitian structure on a 4-manifold M , i.e. a riemannian metric g and two g-compatible complex structures J ± with J + (x) = ±J -(x) at some point x ∈ M , and such that J + and J -induce the same orientation on M . Notice that any riemannian metric which is conformal to g is again bihermitian with respect to J ± ; we can therefore define a bihermitian conformal structure (c = [g], J + , J -) on M .

For a fixed metric g ∈ c, we consider the fundamental 2-forms F g + and F g -of the hermitian metrics (g, J + ) and (g, J -), defined by

F g ± (•, •) = g(J ± •, •).
The corresponding Lee 1-forms, θ g + and θ g -, are introduced by the relations dF g

+ = θ g + ∧ F g + and dF g -= θ g -∧F g -, or equivalently θ g ± = J ± (δ g F g ± )
, where δ g is the co-differential of g and the action of an almost complex structure J on the cotangent bundle T * (M ) is given by (Jα)(X) = -α(JX). Notice that with respect to a conformal change of the metric g = e f g, the Lee forms change by θ g ± = θ g ± + df . The following result was established in [START_REF] Apostolov | The Riemannian Goldberg-Sachs Theorem[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]. Lemma 1. For any bihermitian metric (g, J + , J -), the Lee forms θ g ± satisfy

2δ g θ g + + |θ g + | 2 g = 2δ g θ g -+ |θ g -| 2
g , where | • | g is the point-wise norm induced by g.

Moreover, the 2-form d(θ g + + θ g -) is anti-selfdual so that, when M is compact, θ g + + θ g -is closed. The complex structures J ± must satisfy the relation [START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF] (1)

J + J -+ J -J + = -2p Id,
where p = -1 4 trace(J + J -) is the so-called angle function which, at each point x ∈ M , verifies |p(x)| ≤ 1 with p(x) = ±1 if and only if J + (x) = ±J -(x). In particular, in the strongly bihermitian case, |p| < 1 everywhere.

Another natural object associated to the bihermitian structure (g, J + , J -) is the g-skew-symmetric endomorphism [J + , J -] = J + J --J -J + which anti-commutes with both J ± . Using the metric g, we define a real J ± -anti-invariant 2-form by

Φ g (•, •) = 1 2 g([J + , J -]•, •). Letting Ψ g ± (•, •) := -Φ g (J ± •, •), we consider the complex 2-forms Ω g ± (•, •) = Φ g + iΨ g
± , which are of type (2, 0) on the respective complex manifolds (M, J ± ).

Using [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF] and that F g ± , Φ g , Ψ g ± are all selfdual 2-forms satisfying the obvious orthogonality relations given by their types with respect to J ± , it is straightforward to check (see [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Lemma 2])

F g + = pF g -+ Ψ g -, F g -= pF g + -Ψ g + , Φ g ∧ Φ g = Ψ g ± ∧ Ψ g ± = 2(1 -p 2 )dv g , Φ g ∧ Ψ g ± = 0, Ψ g + ∧ Ψ g -= pΦ g ∧ Φ g , (Ψ g -) 1,1 J + = (1 -p 2 )F g + , (2)
where (•) 1,1 J + denotes the J + -invariant part of a two form. Still using the metric g, we define the g-duals of the (0, 2)-forms Ωg ± , which are smooth sections, say σ g ± , of the respective anti-canonical bundles K -1

J ± = ∧ 2 (T 1,0 J ± (M ))
. Notice that at any point where

J + (x) = ±J -(x), Ω g
± and σ g ± do not vanish, because of (1).

The following key result was established in [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Lemmas 2,[START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF].

Lemma 2. For any bihermitian metric (g, J + , J -) the (2, 0)-forms Ω g ± and bi-vector fields σ g ± satisfy

dΩ g ± = 1 2 (θ g + + θ g -) + d log(1 -p 2 ) ∧ Ω g ± , ∂± σ g ± = - 1 2 (θ g + + θ g -) 0,1 J ± ⊗ σ g ± ,
where the first equality holds on the open subset of points where

J + (x) = ±J -(x),
and in the second equality ∂± stand for the corresponding Cauchy-Riemann operators on the respective anti-canonical bundles K -1 J ± , and (•) 0,1 J ± denote the (0, 1)-parts taken relatively to the respective complex structure.

The first equality in the above lemma tells us that the real 2-forms Φ g , Ψ g ± satisfy

dΦ g = τ ∧ Φ g , dΨ g ± = τ ∧ Ψ g ± with τ = 1 2 (θ g + + θ g -) + d log(1 -p 2
). Together with the relations (2), this allows to reconstruct the strongly bihermitian metric from the 2-forms Φ g , Ψ g + , Ψ g -. Proposition 1. Suppose Φ, Ψ + , Ψ -are non-degenerate real 2-forms on M which satisfy the relations

Φ 2 = Ψ 2 + = Ψ 2 -, Φ ∧ Ψ ± = 0, dΦ = τ ∧ Φ, dΨ ± = τ ∧ Ψ ± ,
for some 1-form τ .

Then, there exists a strongly bihermitian metric (g,

J + , J -) on M with Φ g = Φ, Ψ g ± = Ψ ± if and only if Ψ + ∧ Ψ -= pΦ 2
, where p is a smooth function with |p| < 1

Proof. The necessity of the condition follows from (2) and Lemma 2. (Recall that p(x) = ±1 if and only if

J + (x) = ±J -(x)).
In the other direction the result was originally established in [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Thm. 2] in the case when τ = 0. Following [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF], the almost complex structures J ± are introduced by Ψ ± (•, •) = -Φ(J ± •, •), while the conformal structure is determined by the property that Φ, Ψ + , Ψ -are selfdual 2-forms. The only notable difference with the proof given in [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Thm. 2] is in establishing the integrability of J ± . For that we would need to generalize [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Lemma 5] in order to show that for any pair of 2-forms, (Φ, Ψ), satisfying Φ 2 = Ψ 2 , Φ ∧ Ψ = 0, the almost-complex structure they define by Ψ(•,

•) = -Φ(J•, •) is integrable, pro- vided that dΦ = τ ∧ Φ, dΨ = τ ∧ Ψ,
for some 1-form τ . This fact is well-known (see e.g. [START_REF] Salamon | Special structures on 4-manifolds[END_REF]) but we give here a short proof for completeness. Note that, by the assumption made, the differential of the (2, 0)-form Ω = Φ + iΨ does not have a component of type [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF][START_REF] Apostolov | The Riemannian Goldberg-Sachs Theorem[END_REF]. Thus, for any two complex vector fields V , W of type (0, 1), and any complex vector field U of type (1, 0), we have

0 = dΩ(U, V , W ) = -Ω([ V , W ] 1,0 , U ).
As Ω is a non-degenerate 2-form on the complex vector bundle T 1,0 J (M ), it follows that [ V , W ] 1,0 = 0, i.e. J is integrable.

In the case when J + = J is given, there is a useful ramification of the above criterion, due to Hitchin.

Corollary 1.

[20] Let (M, J) be a complex surface endowed with a non-vanishing complex (2, 0)-form Ω = Φ + iΨ such that dΩ = τ ∧ Ω for some real 1-form τ .

Then, any real 2-form Ψ -on M satisfying Ψ 2 -= Φ 2 , dΨ -= τ ∧ Ψ -, and whose (1, 1)-part with respect to J is positive definite, gives rise to a strongly bihermitian metric (g, J + , J -) on M with J + = J.

Proof. Let F := (Ψ -) 1,1
J be the positive definite (1, 1)-component of Ψ -with respect to the complex structure J. By putting g(•, •) = F (•, J•), we obtain a Riemannian metric with respect to which Φ, Ψ + = Ψ and Ψ -are selfdual 2-forms of equal length. It then follows that Ψ + ∧ Ψ -= pΦ ∧ Φ for a smooth function p satisfying |p(x)| ≤ 1 with p(x) = ±1 if and only if Ψ + (x) = ±Ψ -(x). The later inequality is impossible because the (1, 1)-part of Ψ -is positive definite while Ψ + is J-anti-invariant. (This is also consistent with the last identity in (2).) We can now apply Proposition 1.

On a compact bihermitian 4-manifold (M, g, J + , J -), an overall assumption which we will make from now on, Lemma 2 has the following interpretation.

To simplify notation, consider one of the complex structures, say J = J + , and drop the index + for the corresponding (2, 0)-form Ω g , bi-vector field σ g , canonical bundle K J , Cauchy-Riemann operator ∂, etc.

Since 1 2 (θ g + +θ g -) is a closed 1-form by Lemma 1, it defines a holomorphic structure on the trivial (smooth) complex bundle M × C, by introducing the connection

∇ := ∇ 0 + i 2 J(θ g + + θ g -)
, where ∇ 0 is the flat connection on M × C; the new holomorphic structure depends, in fact, only on the de Rham class a = 1 2 [θ g + +θ g -], so we denote this holomorphic line bundle by L a . This is consistent with the sequence of natural morphisms

(3) H 1 (M, R) → H 1 (M, R * + ) ֒→ H 1 (M, C * ) → H 1 0 (M, O * ),
where the first morphism is induced by the exponential map (and hence is an isomorphism), and H 1 0 (M, O * ) is the space of equivalent classes of topologically trivial holomorphic bundles. We will say that a topologically trivial holomorphic line bundle

L ∈ H 1 0 (M, O * ) is of real type if it is in the image of H 1 (M, R * + ).
The second equality of Lemma 2 simply means that σ g is a holomorphic section of K -1 J ⊗ L -a , while the first equality and (2) imply that its inverse, 1

|σ g | 2 g
Ω g , is a meromorphic section of K J ⊗ L a (where L -a = L -1 a in terms of the morphisms (3)). Notice that by its very definition, σ g (x) = 0 if and only if J + (x) and J -(x) commute, which in view of (1), means J + (x) = ±J -(x). In other words, K -1 J ⊗ L -a ∼ = O if and only if (c, J + , J -) is strongly bihermitian.

We now recall the definition [START_REF] Gauduchon | Le théorème de dualité pluriharmonique[END_REF] of degree of a holomorphic line bundle with respect to a standard hermitian metric on (M, J). A hermitian metric g on a compact complex surface (M, J) is called standard if its fundamental 2-form F g is dd c -closed (equivalently ∂ ∂F g = 0); this is the same as requiring that the corresponding Lee form θ g is co-closed. A fundamental result of Gauduchon (see e.g. [START_REF] Gauduchon | La 1-forme de torsion d'une variété hermitienne compacte[END_REF]) affirms that such a metric exists, and is unique up to homothety, in each conformal class c of hermitian metrics on (M, J). Given a standard metric g on (M, J), one defines the degree of a holomorphic line bundle L by [START_REF] Apostolov | Generalized Kähler manifolds, Commuting Complex Structures, and Split Tangent Bundles[END_REF] deg

g (L) = 1 2π M ρ ∧ F g ,
where ρ is the real curvature form of any holomorphic connection on L (with the usual convention that ρ represents 2πc 1 (L) R ). Thus defined, it has the usual properties of degree, notably it gives the volume with respect to g of the divisor defined by any meromorphic section of L; in particular, the degree is non-negative for bundles with holomorphic sections, and is positive if there are sections with zeroes. For a topologically trivial holomorphic line bundle of real type, say L a (a ∈ H 1 dR (M, R)), its degree with respect to a standard metric g is easy to compute: we choose a representative closed 1-form ξ for a and consider the holomorphic connection ∇ ξ = ∇ 0 + iJξ on L a , with curvature ρ = d c ξ. Substituting in (4) and integrating by parts gives [START_REF] Barth | Compact Complex Surfaces[END_REF] deg

g (L a ) = 1 2π ξ, θ g g = 1 2π a g h , θ g h g ,
where a g h is the harmonic representatives of a with respect of g, θ g h is the harmonic part of the Lee form θ g and •, • g is the global L 2 product on 1-forms induced by the standard metric g. (For the last equality we have used that θ g is co-closed.)

In the case of a bihermitian conformal structure (c, J + , J -) on (M, J) (with J = J + ), we take a standard metric g in c with respect to J and calculate the degree of L a with a = 1 2 [θ g + + θ g -] as above. By using the first relation of Lemma 1 one gets (see [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF])

(6) deg g (L a ) = 1 4π θ g + + θ g -, θ g + g = 1 8π ||θ g + + θ g -|| 2 L 2 .
Notice that by [START_REF] Gauduchon | La 1-forme de torsion d'une variété hermitienne compacte[END_REF], the harmonic part of θ g h is zero if b 1 (M ) is even. In this case the degree of L a must be zero, hence L a = O and θ g + + θ g -= 0 with respect to the standard metric of (c, J + ) (which is therefore standard too with respect to J -), cf. [3, Lemma 4].

As σ g is a non-zero holomorphic section of K -1 J ⊗ L -a , this bundle has always positive or zero degree (the later is possible only when the bundle is trivial, i.e. the metric is strongly bihermitian). We thus obtain that deg g (K -1 J ) is either positive or is zero (and this is true for any standard metric on (M, J)). The zero case can only happen if both K J and L are trivial, and θ g + + θ g -= 0 with respect to the standard metric g of the bihermitian conformal structure. However, the later combination is impossible when b 1 (M ) is odd, as shown in [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Prop. 4].

We summarize our discussion in the following proposition which gathers most of the information used so far to narrow the list of compact complex surfaces which can possibly admit bihermitian metrics [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF][START_REF] Dloussky | On surfaces of class VII + 0 with numerically anticanonical divisor[END_REF][START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF]. Proposition 2. [START_REF] Apostolov | Bihermitian surfaces with odd first Betti number[END_REF][START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF] Let (M, J) be a compact complex surface. Any bihermitian conformal structure (c, J + , J -) with J = J + defines a topologically trivial holomorphic line bundle L of real type and non-positive degree with respect to the standard metric of (c, J), and a non-zero holomorphic section of K -1 J ⊗ L. In particular, the degree of K -1

J must be non-negative with respect to the standard metric of (c, J). Moreover, L is trivial when b 1 (M ) is even and the degree of K -1

J is positive when b 1 (M ) is odd.

The holomorphic bundle K -1 J ⊗ L is trivial if and only if (g, J + , J -) is strongly bihermitian.

Remark 1. One can show that on a compact bihermitian surface the signs of deg g (L) and deg g (K) are the same with respect to any standard metric on (M, J).

In the case when b 1 (M ) is even this is obvious because, by Proposition 2, L = O and H 0 (M, K -1 ) = 0.

In the case when b 1 (M ) is odd, Proposition 2 tells us that the degree of the anti-canonical bundle (with respect to a particular standard metric) is positive and, therefore, by the properties of degree, the pluricanonical line bundles K ⊗m J (m ≥ 1) do not have non-zero holomorphic sections, i.e. (M, J) is of Kodaira dimension -∞. This means that (M, J) belongs to the class VII of the Enriques-Kodaira classification [START_REF] Barth | Compact Complex Surfaces[END_REF]. In particular, b 1 (M ) = 1. According to [START_REF] Gauduchon | Le premier espace de cohomologie de deRham d'une surface complexe à premier nombre de Betti impair[END_REF] 1 , on a compact complex surface with odd first Betti number the harmonic part of the Lee form θ g h of any standard metric g is non-zero. Since the space of standard metrics is convex (and b 1 (M ) = 1), we obtain from (5) that deg g (L a ) has the same sign (positive or zero by ( 6)) for any standard metric g on (M, J). The same is true for deg g (K -1 ) because, by Proposition 2, H 0 (M, K -1 ⊗ L) = 0. Remark 2. A bihermitian metric (g, J + , J -) corresponds to a (twisted) generalized Kähler structure on T M ⊕ T * M in the sense of [START_REF] Gualtieri | Generalized complex geometry[END_REF] if it verifies the extra relations

d c + F g + + d c -F g -= 0, dd c ± F g ± = 0
. Since J ± yield the same orientation on M , the first relation is equivalent (by applying the Hodge * operator) to θ g + + θ g -= 0, while the second relations mean that g is standard metric with respect to both J ± (i.e. δ g θ g ± = 0). Thus, for a bihermitian metric to come from a generalized Kähler structure the line bundle L in Proposition 2 must be trivial. Conversely, any bihermitian conformal structure (c, J + , J -) with L = O satisfies the above conditions with respect to the standard metric of (c, J + ) (and thus, by [START_REF] Gualtieri | Generalized complex geometry[END_REF]Thm. 6.37], induces a twisted generalized Kähler structure on T M ⊕ T * M ). This follows from the formula (6) which shows that θ g + + θ g -= 0 with respect to the standard metric of (c, J + ), provided that deg g (L) = 0. Proposition 2 implies that a strongly bihermitian surface with odd first Betti number is never of this type.

Strongly bihermitian metrics on complex surfaces with odd first

Betti number-Proof of Theorem 1

We now specialize to the case when (M, J) has odd first Betti number and admits a strongly bihermitian metric.

As we explained in Remark 1, the fact that (M, J) has a standard metric with respect to which the degree of the anti-canonical bundle is positive (see Proposition 2) implies that (M, J) belongs to the class VII of the Enriques-Kodaira classification [START_REF] Barth | Compact Complex Surfaces[END_REF]. In particular, b 1 (M ) = 1 and b + (M ) = 0, where b

+ (M ) = 1 2 (b 2 (M )+σ(M )
) is the dimension of the space of self-dual harmonic 2-forms with respect to any metric on M , and σ(M ) is the topological signature. In the strongly bihermitian case K J = L is topologically trivial (see Proposition 2), and then

0 = c 2 1 (M ) = 2e(M ) + 3σ(M ) = -b 2 (M )
, where e(M ) = 2 -2b 1 (M ) + b 2 (M ) is the Euler characteristic. Thus, (M, J) is a minimal surface of class VII with zero second Betti number. By Bogomolov's theorem (see [START_REF] Teleman | Projectively-flat surfaces and Bogomolov's theorem on class VII0 surfaces[END_REF] or [START_REF] Li | On projectively-flat hermitian manifolds[END_REF] for a proof), (M, J) is either a minimal Hopf surface or an Inoue surface described in [START_REF] Inoue | On surfaces of class VII0[END_REF].

Combining Proposition 2 with a result of [START_REF] Teleman | The pseudo-effective cone of a non-Kählerian surface and applications[END_REF]Rem. 4.2] allows us to exclude the case of Inoue surfaces. We reproduce the proof for completeness. Lemma 3. [START_REF] Teleman | The pseudo-effective cone of a non-Kählerian surface and applications[END_REF] The degree of the anticanonical bundle of an Inoue complex surface with zero second Betti number is negative with respect to any standard metric. By Proposition 2, such a complex surface can not admit compatible bihermitian metrics.

Proof. Following [START_REF] Teleman | The pseudo-effective cone of a non-Kählerian surface and applications[END_REF], we show that the canonical bundle of an Inoue complex surface has positive degree with respect to any standard metric.

Recall that an Inoue surface is a quotient of H × C (where H denotes the upper half-plane) by properly discontinuous group Γ of affine transformations. An explicit description of all such groups can be found in [START_REF] Inoue | On surfaces of class VII0[END_REF], and they belong to one the following three types.

For the Inoue surfaces S M , the generators of Γ are of the form

γ 0 (w, z) = (αw, βz), γ k (w, z) = (w + a k , z + b k ), k = 1, 2, 3, with α, a k ∈ R and α|β| 2 = 1 (where w ∈ H, z ∈ C). The tensor Im(w) ∂ ∂w ∧ ∂ ∂z ⊗ ∂ ∂ w ∧ ∂ ∂ z is clearly invariant under Γ,
and therefore defines (after symmetrising) an hermitian metric on the canonical bundle of S M . The curvature of this metric is -i∂ ∂ log(Im(w)) = i Im(w) 2 dw ∧ d w, which is a non-negative (but non-zero) 2-form, so the degree of the canonical bundle is positive with respect to any standard metric.

Similarly, for the other two types of Inoue surfaces, S + N,p,q,r,t and S - P,p,q,r , the generators of Γ are of the form

γ 0 (w, z) = (αw, ǫz + t), γ k (w, z) = (w + a k , z + b k w + c k ), k = 1, 2, 3, with α, a k , b k , c k ∈ R and ǫ = ±1. Now a Γ-invariant tensor is Im(w) 2 ∂ ∂w ∧ ∂ ∂z ⊗ ∂ ∂ w ∧ ∂ ∂ z
which defines an hermitian metric on the canonical bundle with non-negative curvature 2i Im(w) 2 dw ∧ d w; one concludes as before that the degree of the canonical bundle is positive with respect to any standard metric.

Let us now examine the Hopf surfaces. These are, by definition, compact complex surfaces with universal covering space C 2 \ {(0, 0)}. It was shown by Kodaira [24] that the fundamental group Γ of such a surface is a finite extension of the infinite cyclic group Z. The list of concrete realizations of Γ as a group of automorphisms of C 2 can be found in [START_REF] Kato | Topology of Hopf surfaces[END_REF] and we shall make extensive use of this classification in the following rough form: Γ ∼ = γ 0 ⋉ H, where • γ 0 denotes the infinite cyclic group generated by a contraction of C 2 of the form

(7) γ 0 (z 1 , z 2 ) = (αz 1 + λz m 2 , βz 2 ), where α, β, λ ∈ C with 0 < |α| ≤ |β| < 1, λ(α -β m ) = 0, m ∈ N * ; • H is a finite subgroup of U (2), subject to the following constraint (i) if λ = 0, then H ⊂ U (1) × U (1) is abelian and commutes with γ 0 ; (ii) if λ = 0 and |α| = |β|, then H ⊂ U (1) × U (1).
Lemma 4. The Hopf surfaces with canonical bundle of real type are precisely those described in Theorem 1. In particular, by Proposition 2, these are the only Hopf surfaces which can possibly admit a compatible strongly bihermitian metric.

Proof. This is rather standard (see e.g. [9, p.669]). On a Hopf surface (M, J), the morphism

H 1 (M, C * ) → H 1 0 (M, O *
) is injective (it is also surjective [24, II,p.699]). This is because flat C * -bundles are in bijection with C * -representations of Γ = π 1 (M ) at one hand, and a non-vanishing holomorphic function f on C 2 , which satisfies f • γ 0 = αf, α ∈ C * (where γ 0 is the contraction (7)) must be constant, at the other. It follows that a flat C * -bundle is of real type if and only if the corresponding C * -representation of Γ takes values in R * + . The pull-back of the canonical bundle K to the universal covering space C 2 \ {(0, 0)} is trivialized by the holomorphic (2, 0)-form Ω = dz 1 ∧ dz 2 . The contraction γ 0 acts by γ * 0 (Ω) = (αβ)Ω, while for any element h of the finite group H ⊂ U (2) we have h * (Ω) = det(h)Ω. It follows that K -1 is a flat C * -bundle associated to the representation γ 0 → αβ, h → det(h); it is therefore of real type if and only if αβ ∈ R * + and H ⊂ SU (2). From this and Kato's classification mentioned above, the cases (a), (b) and (c) of Theorem 1 follow easily: The particular form of the contraction γ 0 is obtained from [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF] by putting αβ = a ∈ R * + . There is nothing more to prove about H in the case (a) (which corresponds to |α| = |β| in Kato's classification). For the cases (b) and (c) we have |α| = |β|, λ = 0 and λ = 0, respectively. From Kato's classification H ⊂ S(U (1) × U (1)). As H ⊂ SU (2), it acts freely on the unit sphere S 3 ⊂ C 2 (because the action of SU (2) is the same as S 3 acting on itself by left multiplications). It follows that H acts freely on S 1 × {0} and {0} × S 1 . Thus, the projections of U (1) × U (1) to its factors inject H ⊂ U (1) × U (1) into S 1 . Since H is finite, it must be cyclic. There is a unique finite cyclic sub-group of S(U (1)×U ( 1)) of order ℓ, namely the group generated by (z 1 , z 2 ) → (εz 1 , ε -1 z 2 ) where ε is a primitive ℓ-th root of 1.

Finally, as the case (c) corresponds to λ = 0 in Kato's classification, Γ must be abelian, meaning that (z 1 , z 2 ) → (εz 1 , ε -1 z 2 ) (with ε primitive ℓ-th root of 1) commutes with γ 0 . This places the constraint m = kℓ -1, k ∈ N * .

We now turn to the construction of strongly bihermitian metrics on the Hopf surfaces listed in Theorem 1.

Proposition 3. Any Hopf surface (M, J) described in Theorem 1 admits a strongly bihermitian metric (g, J + , J -) with J + = J.

Proof. The Hopf surfaces described in the case (a) of the theorem are hyperhermitian and the complex structure J belongs to the underlying hypercomplex family (cf. [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF][START_REF] Pontecorvo | Complex structures on Riemannian four-manifolds[END_REF]). To see this directly, notice that in this case the fundamental group respects the standard (flat) hyperhermitian conformal structure on C 2 . Notice also that any hyperhermitian metric on a compact complex surface can be deformed to obtain non-hyperhermitian strongly bihermitian metrics [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF]Prop. 1].

Let us now consider the Hopf surfaces described in the case (b) of Theorem 1. Recall that in terms of the Kato classification referred to earlier, this is the case when λ = 0 and αβ = a ∈ R * + in [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF], and H is a finite cyclic subgroup of S(U (1) × U (1)). We shall use Corollary 1 (with τ = 0) first on the universal covering space

C 2 \ {(0, 0)}, then (with τ = 0) on the quotient (M, J) = C 2 \ {(0, 0)} /Γ.
Following [START_REF] Hitchin | Bihermitian metrics on Del Pezzo Surfaces[END_REF]Prop. 3], consider the closed (2, 0)-form Ω = dz 1 ∧ dz 2 = Φ + iΨ on C 2 \ {(0, 0)} and a function f which is a Kähler potential (i.e. dd c f > 0). Let X be the Φ-hamiltonian vector field corresponding to f , defined by i X Φ = df . Put Ψ t -:= φ * t (Ψ), where φ t is the flow of X. Then the forms (Φ, Ψ, Ψ t -) satisfy the conditions of Corollary 1 (with τ = 0), except the last (positivity) condition. However, we have (Ψ 0 -) 1,1 = 0 and

(8) ∂ ∂t Ψ t -t=0 = L X Ψ = d(i X Ψ) = -d(i JX Φ) = dd c f, so (Ψ t -) 1,
1 is positive definite for small t (at least in a neighborhood of each point). We now have to make a special choice of the Kähler potential f so that the construction descends to the quotient of C 2 \ {(0, 0)} by Γ. In fact, we shall use the Kähler potentials introduced by Gauduchon-Ornea [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF] in order to construct (explicit) locally conformally Kähler metrics on Hopf surfaces. Following [14, Remark 2], for any α, β (as in [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF] with λ = 0) consider the flow ϕ t (z 1 , z 2 ) = (α t z 1 , β t z 2 ) of a vector field generating the contraction, i.e. γ 0 = ϕ 1 (this requires a choice of arg(α) and arg(β)). We then define a function r α,β : C 2 \ {(0, 0)} → R as follows: r α,β (z) is the real number such that ϕ -r α,β (z) belongs to the unit sphere S 3 ⊂ C 2 . It is shown in [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF] that this definition is correct and that the positive real function f α,β = exp (ln |α| + ln |β|)r α,β is a Kähler potential on C 2 \ {(0, 0)}. In fact, as r α,β (γ 0 • z) = r α,β (z) + 1 by definition, we have

(9) f α,β (γ 0 • z) = |α||β|f α,β (z),
By the explicit formulae in [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF], f α,β is also U (1) × U (1)-invariant (and is U (2)invariant when |α| = |β|). It follows that

F α,β := 1 f α,β dd c f α,β
is a Γ-invariant positive definite (1, 1)-form which defines a locally conformally Kähler metric on the corresponding Hopf surface (M, J). We use the function f α,β in our construction of the 2-forms (Φ, Ψ, Ψ t -) as above. First of all, because in our case αβ = |α||β| and H ⊂ S(U (1)× U (1)), it follows from (9) that the corresponding vector field X is Γ-invariant, and thus X and its flow φ t are defined on (M, J). Secondly, by ( 9), the 2-forms ( 1 f α,β Φ, 1 f α,β Ψ, 1 f α,β Ψ t -) are Γinvariant, and therefore define a triple of 2-forms ( Φ, Ψ, Ψt -) on (M, J), satisfying the conditions of Corollary 1 (with τ = -d log f α,β ), except possibly the last (positivity) condition. But as f α,β is invariant under φ t by construction, we get by ( 8)

∂ ∂t Ψt -t=0 = F α,β ,
which is a positive definite (1, 1)-form on the compact manifold (M, J). As the (1, 1)-part of Ψt -at t = 0 is zero, it must be positive definite for all small t different than 0.

To conclude the proof we consider the Hopf surfaces in the case (c) of Theorem 1. These correspond to λ = 0 and α = β m in [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF], with β m+1 = |β| m+1 = a. Recall that in this case H is the cyclic group generated by (z 1 , z 2 ) → (εz 1 , ε -1 z 2 ) with ε primitive ℓ-th root of 1 so that H commutes with the contraction [START_REF] Boyer | A note on hyperhermitian four-manifolds[END_REF].

It is well-known (see e.g. [START_REF] Harvey | An intrinsic characterisation of Kähler manifolds[END_REF][START_REF] Kodaira | Complex manifolds[END_REF]) that any two Hopf surfaces with fundamental groups Γ β,m,λ,ε and Γ β,m,λ ′ ,ε corresponding to generators with the same values of β, m and ε (but possibly different non-zero values λ and λ ′ ) are isomorphic as complex manifolds. Because of this fact, in order to adapt the construction in the case (b) to the case (c), it suffices to find some λ 0 = 0, and a positive function f β,m on C 2 \ {(0, 0)} such that

• f β,m (γ 0 • z) = |β| m+1 f β,m (z), where γ 0 is given by ( 7) with λ = λ 0 ;

• dd c f β,m > 0;

• f β,m is H-invariant.

The argument in [14, p.1125] produces positive functions which satisfy the first two but not, a priori, the third requirement. However, their construction starts by producing a family of positive smooth functions f β,m,λ satisfying the first condition and lim λ→0 f β,m,λ = f β m ,β , where f β m ,β is the function f α,β with α = β m , defined above. Since f β m ,β is H-invariant (it is, in fact, U (1) × U (1)-invariant as one can see from the equation ( 10) in [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF]) and since H commutes with γ 0 , by replacing f β,m,λ with its average over H, we can assume without loss that f β,m,λ are H-invariant. Thus, still following [START_REF] Gauduchon | Locally conformally Kähler metrics on Hopf surfaces[END_REF], the (1, 1)-form F β,m,λ := dd c f β,m,λ f β,m,λ descends to the respective complex manifold (M λ , J λ ). By identifying M λ with M 0 (as smooth manifolds), we have lim λ→0 J λ = J 0 , lim λ→0 F β,m,λ = F β m ,β , where J 0 is the complex structure on M 0 obtained by taking γ 0 with α = β m , λ = 0 (note that H stays unchanged). As F β m ,β is a positive definite (1, 1)-form with respect to J 0 , so is F β,m,λ (with respect to J λ ) for λ sufficiently small.

  Vestislav Apostolov, Département de Mathématiques, UQAM, C.P. 8888, Succ. Centreville, Montréal (Québec), H3C 3P8, Canada, and, Institute of Mathematics and Informatics, Bulgarian Academy of Science, Acad. G. Bonchev Str. Bl. 8, 1113 Sofia, Bulgaria E-mail address: apostolov.vestislav@uqam.ca Georges Dloussky, Centre de Mathématiques et d'Informatique, Laboratoire d'Analyse Topologie et probabilités, Université d'Aix-Marseille 1, 39, rue Joliot-Curie, 13453, Marseille Cedex 13, France E-mail address: dloussky@cmi.univ-mrs.fr

The result used here is readily available in[START_REF] Apostolov | Generalized Kähler manifolds, Commuting Complex Structures, and Split Tangent Bundles[END_REF] Prop. 1].


Charbel Moussa

Olivier Bartier

Xavier Hernot

Gérard Mauvoisin

Jean-Marc Collin
email: jean-marc.collin@univ-nantes.fr

Guillaume Delattre

Mechanical characterization of carbonitrided steel with spherical indentation using the average representative strain

Keywords: Spherical indentation, average representative strain, carbonitrided steels, hardening law, mechanical characterization, confidence domain

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Identification of the mechanical properties of non-homogeneous materials have become an important challenge during the last decades. Knowledge of the hardening laws of these materials is necessary in order to design industrial products. An important material family is the so-called plastically graded materials (PGMs). Among PGMs, carbonitrided steels are used for many industrial applications. These materials present a variation of the hardness with the depth from the surface while the elastic properties remain constant [START_REF] Cao | A new scheme for computational modeling of conical indentation in plastically graded materials[END_REF][START_REF] Garnier | Determining of the evolution of the elasticity modulus by surface wave according to the depth in a nitrided layer[END_REF][START_REF] Giannakopoulos | Indentation of plastically graded substrates by sharp indentors[END_REF][4].

Using classical tensile tests on PGM does not provide a good prediction of the changes of the mechanical properties with the depth. Indeed, the treated volume of material is negligible compared to the volume of the non treated part of a standard tensile sample. A local mechanical test is thus necessary in order to obtain accurate values of the mechanical properties. One of these tests is the instrumented indentation which has been studied for several decades. Its applications in mechanical characterization have been demonstrated in a great number of papers [START_REF] Nayebi | New procedure to determine steel mechanical parameters from the spherical indentation technique[END_REF][START_REF] Collin | Experimental evaluation of the stress-strain curve by continuous indentation using different indenter shapes[END_REF][START_REF] Lee | A numerical approach to spherical indentation techniques for material property evaluation[END_REF][START_REF] Cao | A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve[END_REF][START_REF] Ogasawara | Measuring the plastic properties of bulk materials by indentation test[END_REF][START_REF] Zhao | A new approach to measure the elasticplastic properties of bulk materials using spherical indentation[END_REF][START_REF] Nakamura | Determination of properties of graded materials by inverse analysis and instrumented indentation[END_REF][START_REF] Bolzon | Material model calibration by indentation, imprint mapping and inverse analysis[END_REF][START_REF] Huber | Determination of constitutive properties from spherical indentation data using neural networks. Part II: plasticity with non linear isotropic and kinematic hardening[END_REF][START_REF] Parenteau | Material parameters identification using experimental continuous spherical indentation for cyclic hardening[END_REF][START_REF] Mauvoisin | Materials characterization by instrumented indentation using two different approaches[END_REF]. In most of these papers, the Hollomon's power law given in Eq. [START_REF] Cao | A new scheme for computational modeling of conical indentation in plastically graded materials[END_REF] has been used to model the elastic-plastic behaviour: [START_REF] Cao | A new scheme for computational modeling of conical indentation in plastically graded materials[END_REF] where E is the Young modulus, σ y the yield stress and n the strain hardening exponent.

A great number of papers concern the determination of mechanical properties of homogeneous materials. In this case, the determination of the mechanical parameters of Eq. ( 1) from the indentation curve can be done using the minimization between indentation experiments and models of instrumented indentation obtained by numerical simulations [START_REF] Nayebi | New procedure to determine steel mechanical parameters from the spherical indentation technique[END_REF][START_REF] Collin | Experimental evaluation of the stress-strain curve by continuous indentation using different indenter shapes[END_REF][START_REF] Lee | A numerical approach to spherical indentation techniques for material property evaluation[END_REF][START_REF] Cao | A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve[END_REF][START_REF] Ogasawara | Measuring the plastic properties of bulk materials by indentation test[END_REF][START_REF] Zhao | A new approach to measure the elasticplastic properties of bulk materials using spherical indentation[END_REF][START_REF] Nakamura | Determination of properties of graded materials by inverse analysis and instrumented indentation[END_REF]. The main advantages of this approach are the simplicity of implementation and the instant results. However, the proposed models depend on the behaviour law, the geometry of the indenter and many other assumptions depending on each study. Moreover, the application of these methods is often limited to particular cases of bulk homogeneous materials and for a specific indenter geometry.

In order to apply the indentation test whatever the behaviour law, sample heterogeneity or indenter geometry, a second approach can be used. This approach is called "inverse analysis" where the minimization is directly made between experiments and finite element (FE) simulations. Several authors [START_REF] Bolzon | Material model calibration by indentation, imprint mapping and inverse analysis[END_REF][START_REF] Huber | Determination of constitutive properties from spherical indentation data using neural networks. Part II: plasticity with non linear isotropic and kinematic hardening[END_REF][START_REF] Parenteau | Material parameters identification using experimental continuous spherical indentation for cyclic hardening[END_REF][START_REF] Mauvoisin | Materials characterization by instrumented indentation using two different approaches[END_REF] have proposed to apply the inverse analysis approach on indentation experiments in order to determine some mechanical properties. The two approaches presented above are, for most of them, purely phenomenological. However, a great number of papers have explained the physical phenomenon involved during an indentation test. The indentation test induces a complex tri-axial mechanical behaviour. This is the reason why the concept of representative strain has been introduced by Tabor [START_REF] Tabor | Hardness of Metals[END_REF]. The aim of using the representative strain concept is to make the link between the material response to the indentation test and the material hardening law. For conical indenters, the representative strain is constant whatever the applied load, which explains the non-uniqueness of the solution for the identification of the hardening law [START_REF] Cheng | Can stress strain relationship be obtained from indentation curves using conical and pyramidal indenters?[END_REF][18][19][START_REF] Hernot | Study of the concept of representative strain and constraint factor introduced by Vickers indentation[END_REF]. For conical indentation, all the stress-strain curves that intersect at a point of the abscissa which corresponds to the representative strain, lead to identical indentation curves [START_REF] Hernot | Study of the concept of representative strain and constraint factor introduced by Vickers indentation[END_REF]. For spherical indentation, representative strain depends on the indentation depth. Thus, the problem of uniqueness of the solution can be solved [START_REF] Cheng | Can stress strain relationship be obtained from indentation curves using conical and pyramidal indenters?[END_REF]. The concept of representative strain is still an open question and a great number of formulations and definitions have been proposed [7-8-18-19-20].

In a previous paper, Moussa et al. [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF] proposed a new method to determine the hardening law of homogenous materials using the average representative strain concept. The advantage of this concept is that the representative strain is directly obtained from the response of the indentation test. Using different penetration depths provides the range of strain for which the identified hardening law is valid. Thus, it allows the determination of a confidence domain taking into account experimental imprecision and material heterogeneity. However, in the reference [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF], only bulk homogenous materials were studied.

A complete overview of the theory realized by Moussa et al. [START_REF] Moussa | Characterization of homogenous and plastically graded materials with spherical indentation and inverse analysis[END_REF] has shown that, compared to homogenous materials, few studies have been done to characterize the mechanical behaviour of PGM with the indentation test. They have demonstrated that most of the methods proposed in the literature are based on restrictive assumptions which are not verified for all PGMs. Some of the methods even consider that the strain hardening exponent remains constant between surface and substrate [START_REF] Choi | Mechanics of indentation of plastically graded materials-I: Analysis[END_REF][START_REF] Choi | Mechanics of indentation of plastically graded materials-II: Experiments on nanocrystalline alloys with grain size gradients[END_REF][START_REF] Gao | Two new expanding cavity models for indentation deformations of elastic strain-hardening materials[END_REF], which cannot be physically justified. For those reasons, the authors previously [START_REF] Moussa | Characterization of homogenous and plastically graded materials with spherical indentation and inverse analysis[END_REF] proposed to characterize the mechanical behaviour of two carbonitrided steels using spherical indentation and the "inverse analysis" approach. In this paper, a methodology was proposed, based on the assumption of a linear variation of the hardening law from the surface to the substrate. This assumption has been justified and validated by Moussa et al. [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF].

The aim of this work is to generalize the concept of average representative strain, previously proposed for homogenous materials, to PGMs and to apply this to carbonitrided steels. This concept will be used in order to define a confidence domain around the identified hardening laws. Moreover, a numerical study of the sample behaviour under the indenter will show the non influence of friction.

Characterization method

In previous studies [START_REF] Moussa | Characterization of homogenous and plastically graded materials with spherical indentation and inverse analysis[END_REF][START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF], a procedure to characterize PGMs was presented. This procedure can be described as follows: "The decreasing hardness profile of carbonitrided steels can be approximated by three zones. In the first zone of thickness e 1 , the hardness is constant and equal to the surface hardness. In the second zone of thickness e 2 , the hardness presents a regular decrease and can be approximated by an inclined line. Finally, the third zone corresponds to the substrate (core), where the hardness is constant. The variation of the plastic properties is approximated in the same way" [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF]. This variation, illustrated in Fig. 1, can be described by the following equation Where, for each plastic strain, σ surface and σ substrate are the stress of the surface and of the substrate respectively, z is the distance from surface across the thickness of the sample and e 1 and e 2 are the thicknesses of the surface and intermediate layers respectively. With this approximation, the characterization of the carbonitrided steels is reduced to the identification of the hardening law of the surface, the hardening law of the substrate and the thicknesses e 1 and e 2 . The present procedure is divided into the three steps illustrated in Fig. 1. The first step consists of applying a spherical indentation test to the substrate. Since the volume of the substrate is large enough, it can be considered as a homogenous material and the material parameters set ( ysubstrate , n substrate ) of the hardening law of the substrate material can be identified. The second step consists of estimating the values of thicknesses e 1 and e 2 from the hardness profile. The third step consists of applying a spherical indentation test on the surface. Since the hardening laws of the substrate, e 1 and e 2 are identified, the only unknown values left to identify are the material parameters set ( ysurface , n surface ) of the hardening law of the surface. Following the proposed procedure, the problem is divided into three steps with two unknowns for each.

The described three steps approach was presented previously. [START_REF] Moussa | Characterization of homogenous and plastically graded materials with spherical indentation and inverse analysis[END_REF][START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF] and applied with inverse analysis using SiDoLo software [27]. In the present paper, these three steps of characterization were performed using the identification method proposed [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF] for homogeneous materials. The simulation of the spherical indentation test was performed with the FE method using the commercial FE software ABAQUS. Simulations were done using an axisymmetric mesh built with 18,400 elements. The hardening law used in this paper is the Hollomon's power law given in Eq. ( 1) associated to a Von-Mises plasticity criterion.

Application of a characterization methodology based on the average representative strain. [21]

Material and experimental parameters

In this paper, the studied material is the C12-carbonitrided steel for which the hardening profile and metallographic analysis have been presented in a previous paper [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF]. Before performing the carbonitriding process, the specimens were carefully polished using fine emery papers (up to 1200 grit) to avoid uncertainties due to roughness when performing the indentation test on the surface of the carbonitrided steel. The carbonitrided specimens were cut out and the sections were polished using fine emery papers (up to 1200 grit) and diamond suspensions (6 and 3 μm) before performing the Vickers micro hardness tests and the metallographic analysis. Vickers micro hardness profile of the transverse cross section of the

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carbonitrided steel using a 200g load provided the values for the thicknesses e 1 =0.06mm and e 2 =0.29mm [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF].

The metallographic analysis showed that the microstructure of the carbonitrided layers consists of a martensite or bainite matrix containing troostite particles in the former austenite grain boundaries. This analysis also shows that the microstructure of the substrate mainly contains acicular ferrite and pearlite grains. The size of the former austenitic grains is smaller than 50μm. The diagonal lengths of the Vickers imprints in the range from 20μm to 40μm lead to hardness of the order of 700HV and 220HV for the surface and the substrate of the carbonitrided steel, respectively [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF].

The spherical indentation tests were carried out using an indentation instrumented bench developed within our laboratory. The system's resolutions are 0.02N and 0.02μm for load and displacement respectively. The tests were carried out with a spherical indenter of radius R 0.5 mm, and the displacement rate was about 2μm/s. The maximum applied load was equal to 900 N. The diameter of the residual imprint performed on the substrate after applying 900N was about 730μm. Compared to the grain size of the microstructure, the size of the imprint is large enough to avoid structural heterogeneity problems. Concerning the spherical indentation test performed on the surface of the carbonitrided steel, the plastic zone depth is larger than the thickness of the carbonitrided layers when applying a maximum applied load of 900N. This indicates that the response of the indentation test is influenced by the surface, the intermediate layers and the substrate.

Characterization of the substrate

In previous studies [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF] a characterization method with spherical indentation for homogenous materials was proposed. This method was used to identify the hardening law of the substrate of the C12-carbonitrided steel. The fundamental concepts used in order to determine the mechanical properties of the substrate fully illustrated in previous papers [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF] are given in the following paragraph.

The method consists of calculating the error between the average experimental F-h curve (average load for every penetration depth) and a number of F-h curves obtained from FE simulation. In order to quantify the difference between two indentation curves, the root mean square error, Eq. 3, was used:

dh F F h R h E h RMS 2 0 2 1 max max max 1 (3)
Where R is the spherical indenter radius (R= 0.5mm), h is the penetration depth, h max is the maximal penetration depth and F 1 and F 2 are the load for the two considered curves. To use this method, a database of FE simulation indentation curves was built up with 3,872 F-h curves. The corresponding Hollomon hardening law parameter sets ( y , n) were presented in a previous paper [START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF]. Six indentation tests were done on three different samples of the carbonitrided C12 steel. The average curve was only used to identify the substrate of the carbonitrided C12 steel. The characterization procedure consists of calculating the difference, using E RMS (Eq. 3), between the average experimental F-h curve and the F-h curves of the database. The presence of a valley, in which small variation of E RMS occurs, was shown in our

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previous paper [START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF]. This distribution takes the form of a cone with an elliptical base in the [k, n] diagram with k equal to:

E n k y ln 1 (4)
It should be noticed that the elliptical cone form was assumed to be the form of the ERMS distribution near the solution, i.e where the valley exists. For this reason, E critical was defined in order to use only the material parameter sets that are near the solution to correctly determine the parameters of the elliptical cone. In this study E critical was set to 30N. The coordinates of the elliptical cone summit, corresponding to the identified solution and the direction of the principal axis of the ellipse in [k, n] the diagram, indicates the average representative strain aR . From this procedure, the material parameter set (σ y , n) that represents the solution was identified and the average representative strain aR was determined. σ y =315 Mpa and n=0.166 corresponding to aR = 0.054 were identified for the substrate. In order to take into account the material heterogeneity and the experimental imprecision, the gap between the average curve and each one of the experimental curves was quantified using ERMS, Eq. 3. For the case of the substrate of the carbonitrided C12 steel, the maximal value of experimental gap is equal to 7.3N. The isovalue of 7.3N was used to determine the confidence domain [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF]. All the Hollomon hardening laws that are located inside the confidence domain will lead to F-h curves that give an ERMS value with the average experimental F-h curve smaller than 7.3N. The identified hardening law, (σ y =315Mpa, n=0.166) and the confidence domain are presented in Fig. 2. The confidence domain is narrow because the experimental F-h curves are very close. As mentioned above, the thicknesses e 1 and e 2 were estimated from the hardness profile. For the studied C12 carbonitrided steel, e 1 =0.06mm and e 2 =0.29mm [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF]. At this stage, since the material parameter set (σ y , n) of the substrate was identified, the two unknowns are the material parameter sets (σ y , n) of the surface. In a previous study [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF], the identification of the material parameter set (σ y , n) of the surface was calculated with inverse analysis. In the present study, this identification was done using the method that was used for the substrate (paragraph 3.1). In order to build up a database of FE simulation F-h curves of the surface, the material parameter set identified for the substrate,

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(σ y =315 Mpa, n=0.166) and the thicknesses, e 1 =0.06mm and e 2 =0.35mm were fixed. The material parameters sets chosen to build up the database are presented in Table1.

The F-h curves obtained on the surface of three samples are presented in Fig. 3. As we can see, for the case of the surface of the carbonitrided C12 steel the reproducibility of the curves is not as good as for the case of the substrate. This observation was expected since when the indentation test was done on the surface, the intermediate zone and the substrate influence the behaviour of the material. Therefore the F-h curves of the surface are influenced by the three parts of the material: surface, intermediate layer and substrate. The differences observed between the three samples for each one of these three layers and for the hardness profile explain the differences observed in the F-h curves. The average curve of the nine experimental curves obtained on the surface was used to identify the surface material parameters. The ERMS was calculated between the average experimental curve and each one of the F-h curves of the database. The ERMS distribution obtained is presented in Figs. 45. This observation means that the previous identification method (based on the cone with the elliptical base) proposed for the homogenous material can be applied for the case of PGMs. Figs. 4-5 also show that the valley is much larger than the one obtained for homogenous materials [START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF]. This observation means that the sensitivity of the F-h curve obtained for PGM is smaller than that of the F-h curve obtained for homogenous material. This result was

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somehow expected since the database was built up with a fixed the hardening law of the substrate, which means that the sensitivity of only a part of the plastified volume under the indenter is tested.

The comparison between the ERMS distribution obtained from the database and the one obtained from the equations of the cone with the elliptical base is presented in Fig. 6. In this figure, it is clearly shown that the ellipse distribution superimposes onto the ERMS distribution. The result given in this figure proves that the ERMS distribution takes the form of a cone with an elliptical base. The small differences observed in Fig. 6 are mainly due to the lack of FE simulations in the database.

From the equation of the cone with the elliptical base [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF], the mechanical properties of the surface σ y0 =1916.6MPa and n 0 =0.0961 were determined. The same result was obtained when inverse analysis was used previously. [START_REF] Moussa | Experimental and numerical investigation on carbonitrided steel[END_REF]. tThe results in that study, obtained when using inverse analysis, were validated by using an instrumented roller test. Thus, we can consider that the hardening laws for the carbonitrided steels identified in this paper enable the material response of the C12 carbonitrided steel, under different indentation load conditions, to be predicted.

In the proposed methodology, the third step consists in performing spherical indentation on the hardened surface of the material with the aim of obtaining a plastic zone of thickness higher than the thickness of the hardened layer. From this step, the work hardening law of the hardened layer is obtained. In our opinion, the proposed methodology used with the experimental condition given in the paper, i.e. R=0.5mm, F=900N, can be applied to standard carburized, nitrided or carbonitrided, steels of hardened layer thicknesses smaller than 1.5 mm.

If we want to use the proposed method to accurately characterize thin films of thickness less than 0.05mm, an indenter with a radius smaller than 0.5 mm is required. However, small perfect indenter geometry is quite difficult to manufacture, especially in the nano and micro scales. Indentation curves obtained with imperfect indenter geometry can show great differences to those obtained with assumed perfect indenter geometry, thereby leading to erroneous data exploitation results. In this case, the authors recommend using the correction method proposed in ref. [START_REF] Brammer | A method to take account of the geometrical imperfections of quasi-spherical indenters[END_REF].

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3.3.b. Average representative strain

Since the ERMS distribution takes the form of a cone with an elliptical base, in a same way as for homogenous materials, an average representative strain can be defined for the PGM. The material parameter sets (σ y , n) located at the bottom of the valley give hardening laws that lead to F-h curves which are very close to the F-h experimental curve. Similarly to the case of homogenous materials [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF], all the hardening laws intersect in a same point. We define the "average representative strain" ε ar as the abscissa of the intersection point (Fig. 7). The value ε ar = 0.0615 was analytically determined for the studied case [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF]. All the Hollomon hardening laws that intersect the stress-strain curve corresponding to the solution at [ε ar , σ ar ] lead to indentation curves close to the indentation curve corresponding to the solution. Hence, the F-h curve obtained for a PGM is mostly influenced by the part of the hardening law which is located around ε ar . When a material is characterized using the F-h curve, it is this part of the Hollomon hardening law that is mostly characterized and better identified.

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3.3.c. Solution domain

In the identification method developed for the case of homogenous materials, a confidence domain, which allows the taking into account of experimental imprecision and material heterogeneity, was defined. In this paragraph the confidence domain is determined for the carbonitrided steel. The values of ERMS calculated between the average experimental curve and each one of the nine experimental curves, using Eq. 3, are presented in Table 2. Table 2: Values of the cost function for the nine curves for h max /R = 0.17.

ERMS (N) for

In order to take into account experimental imprecision and material heterogeneity, the maximal value of Table 2 , 16.6 N, was considered. The isovalue corresponding to this value is presented in Fig. 8. In this figure the material parameter set identified for the average curve is presented by a black dot.

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Figure 8: Solution domain for the nine experimental tests for h max /R = 0.17

All the material parameter sets (σ y , n) located inside the domain limited with this isovalue give hardening laws that lead to F-h curves with ERMS values smaller than 16.6N. Therefore all material parameter sets (σ y , n) located inside this domain should be considered as solutions, because this value is the maximal gap between experimental curves. The domain presented in Fig. 8 is defined as the solution domain.

Each one of the nine F-h curves was used to identify the Hollomon hardening law of the surface. The identified parameter sets (σ y , n), see Table 3, are presented with colored dots in Fig. 8. This figure shows that these colored dots are all located inside the solution domain. This result validates the fact that the solution domain regroups all possible solutions.

3.3.d. Confidence domain

The confidence domain, developed in previous studies for the case of homogenous materials [START_REF] Moussa | Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths[END_REF][START_REF] Moussa | Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain[END_REF], regroups all possible solutions in the stress-strain diagram. This domain was obtained from the four points located on the four summits of the solution domain, see Fig. The confidence domain is defined between ε ab and ε bc defined as follow: aR aR cd ab aR aR da bc [START_REF] Collin | Experimental evaluation of the stress-strain curve by continuous indentation using different indenter shapes[END_REF] with X Y aR aR aR arctan ln 1 2 [START_REF] Lee | A numerical approach to spherical indentation techniques for material property evaluation[END_REF] In the present study, we can see in Fig. 9.a that a part of the solution domain corresponds to a negative value of n, which is not realistic for the plasticity of cold metals. Therefore the solution domain should be reduced in order to exclude the hardening laws that have a negative value of n. As we can see in Fig. 9.a, point c should be replaced with points c' and c". The hardening laws of the points a, b, c', c'' and d are presented in Fig. 9.b. As for the average representative strain, the direction of the segment [c'c"] allows an equivalent value of strain to be determined. Since the segment [c'c"] is horizontal, the equivalent strain is infinite.

For the case of the carbonitrided steel, the confidence domain corresponding to the solution domain can be thus determined, see Fig. In order to understand the non-influence of the friction coefficient, the radial displacements obtained from FE simulations are plotted in Fig. 12 for homogenous material, and in Fig. 13 for carbonitrided steel. For the homogenous material, Fig. 12 shows that the highest values of radial displacement are in the contact zone between the indenter and the material. This observation explains why the friction has a strong influence on the spherical indentation response of homogeneous materials. The material tries to move horizontally at the contact zone and the friction prevents this displacement. Therefore, the friction has an influence on the pile up formation and on the F-h curve. This influence has already been demonstrated in various studies. For the carbonitrided steels, it can be observed in Fig. . 13 that there is almost no radial displacement in the contact zone. This means that for friction or frictionless conditions, the material does not move horizontally underneath the indenter which explains the non-influence of the friction coefficient for indentation of carbonitrided steels.

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no radial displacement in the contact zone and also why there is no pile up for the case of carbonitrided steels.

Conclusion

In previous papers, authors have proposed two methods in order to characterize mechanical properties of carbonitrided steels. When mechanical properties are identified, it is very important to have access to the accuracy of these properties.

In this paper, it was demonstrated that using the concept of the average representative strain proposed in a former paper, provides a confidence domain. The average identified hardening law is surrounded with two border functions which represent the confidence domain.

Moreover, a numerical study of the sample behaviour under the indenter has shown the noninfluence of the friction coefficient on the results. In the case of an industrial application, this result demonstrates that identification results will not be perturbed with different friction conditions.

Figure 1 :

 1 Figure 1: Method for the characterization of carbonitrided steel

Figure 2 :

 2 Figure 2: Identified hardening law for the substrate with the confidence domain

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Figure 3 :

 3 Figure 3: Experimental F-h curves on the C12-carbonitrided steel surface

Figure 4 :Figure 5 :

 45 Figure 4 : Isovalues for the nine experimental curves in the (σ y , n) space

Figure 6 :

 6 Figure 6 : Comparison between isovalues calculated with cost functions and the ones calculated with the ellipse equations.

Figure 7 :

 7 Figure 7: Comparison between the identified hardening law and that calculated from the set of parameters which are at the bottom of the valley.

  9.a.Around the representative parameter set [ε ar , σ ar ] the confidence domain is limited by the hardening laws corresponding to points b and d presented in Fig.9.a, determined with the following equations:

  9.b.

Figure 9 .ΘFigure 9

 99 Figure 9.a : Solution domain for the surface

Figure 10 : 4 .

 104 Figure 10: Confidence domain with the nine identified hardening laws from experimental F-h curves.

Figure 11 :

 11 Figure 11 : The three indentation simulated curves F-h with the calculated error between them and the reference μ= 0.1

Figure 12 :Figure 13 :

 1213 Figure 12: Radial displacement h max /R = 0.17 in the case of a homogeneous material ( y = 260 MPa and n = 0,16)

hmax/R=0.17 Sample a Sample b Sample c Test 1 Test 2 Test 3 Test 1 Test 2 Test 3 Test 1 Test 2 Test 3 Maximum

	13.5	7.8	16.6	6.6	2.6	1.5	5.0	14.6	12.9	16.6

Test 2 Test 3 Test 1 Test 2 Test 3 Test 1 Test 2 Test 3

			hmax/R=0.17
		Sample a	Sample b	Sample c	Average curve
	Test 1 σ y (MPa) 1984.6 1901.0 2083.3 1740.0 2047.4 1941.2 2115.8 1840.3 2131.9	1916.6
	n	0.1091 0.1164 0.0927 0.1280 0.0691 0.0901 0.0431 0.0879 0.0190	0.0961

Table 3 :

 3 Identified mechanical properties for the nine experimental curves