Pierre Mounoud

Metrics without isometries are generic

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Let V be a compact manifold and M p,q be the set of smooth pseudo-Riemannian metrics of signature (p, q) on V (we suppose that it is not empty). In [START_REF] Gromov | Lectures on transformation groups: geometry and dynamics, Surveys in Differential Geometry[END_REF] D'Ambra and Gromov wrote:"everybody knows that Is(V, g) = Id for generic pseudo-Riemannian metrics g on V , for dim(V ) ≥ 2." Nevertheless, as far as we know, no proof of this fact is available. The purpose of this short article is to give a proof of this result in the case where V is compact and to precise the meaning of the word generic. Let us recall that it is known since the work of Ebin [START_REF] Ebin | The manifold of Riemannian metrics[END_REF] (see also [START_REF] Kim | Semicontinuity of compact group actions on compact differentiable manifolds[END_REF]) that the set of Riemannian metrics without isometries on a compact manifold is open and dense. We prove:

Theorem 1 If V is a compact manifold such that dim(V ) ≥ 2 then the set G = {g ∈ M p,q | Is(g) = Id} contains a subset that is open and dense in M p,q for the C ∞ -topology.
This result is optimal in the sense that G is not always open as we showed in [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF]. The particularity of the Riemannian case lies in the fact that the natural action of the group of smooth diffeomorphisms of V , denoted by Diff(V ), on M n,0 is proper. Furthermore, Theorem 4.2 of [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF] says that when this action is proper then G is an open subset of M p,q . The idea of proof is therefore to find a big enough subset of M p,q invariant by Diff(V ) on which the action is proper. We have decided to be short rather than self-contained, in particular we are going to use several results from our former work [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF].

In the following M p,q will be endowed with the C ∞ -topology without further mention of it and by a perturbation we will always mean an arbitrary small perturbation.

For any g ∈ M p,q we denote by Scal g its scalar curvature and by M g the maximum of Scal g on V . Let F V be the set of pseudo-Riemannian metrics g such that Scal -1 g (M g ) contains a (non trivial) geodesic. The big set we are looking for is actually the complement of F V .

Proposition 2 The set O V = M p,q F V
is an open dense subset of M p,q invariant by the action of Diff(V ) and the restriction of the action of Diff(V ) to O V is proper.

Proof. The set O V is clearly invariant. Let g ∈ M p,q and x 0 ∈ V realizing the maximum of Scal g . It is easy to find a perturbation with arbitrary small support increasing the value of Scal g (x 0 ). Repeating these deformation on smaller and smaller neighborhood of x 0 we find a perturbation of g such that the maximum of the scalar curvature is realized by only one point (see [START_REF] Ebin | The manifold of Riemannian metrics[END_REF] p. 35 for a similar construction). Hence O V is dense in M p,q . 1 Let us see now that F V is closed. Let g n be a sequence of metrics of F V converging to g ∞ . For any n ∈ N there exists a g n -geodesic γ n such that Scal gn is constant and equal to M gn = max x∈V Scal gn (x) on it. As the sequence of exponential maps converges to the exponential map of g ∞ and as V is compact we see that (up to subsequences) the sequence of geodesics γ n converges to a g ∞ -geodesic γ ∞ . As Scal gn → Scal g∞ we know that Scal g∞ is constant along γ ∞ and its value is necessarily M g∞ . Hence g ∞ ∈ F V and F V is closed.

Let us suppose that the action of Diff(V ) on M p,q is not proper (otherwise there is nothing to prove). It means (see [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF]) that there exists a sequence of metrics (g n ) n∈N converging to g ∞ and a non equicontinuous sequence of diffeomorphisms (φ n ) n∈N such that the sequence of metrics (φ * n g n ) converges to g ′ ∞ . The proposition will follow from the fact that g ∞ or g ′ ∞ have to belong to F V . We first remark that the sequence of linear maps (Dφ n (x n )) n∈N lies in O(p, q) up to conjugacy by a converging sequence. As the sequence (φ n ) n∈N is non equicontinuous, we know by [5, Proposition 2.3] that there exists a subsequence such that Dφ n k (x n k ) → ∞ when k → ∞. We deduce from the KAK decomposition of O(p, q) the existence of what are called in [START_REF] Zeghib | Isometry groups and geodesic foliations of Lorentz manifold. Part I: foundations of Lorentz dynamics[END_REF], see subsection 4.1 therein for details, strongly approximately stable vectors, more explicitly we have: Fact 3 For any sequence (x n ) n∈N of points of V , there exist a sequence (v n ) n∈N such that (up to subsequences):

• ∀n ∈ N, v n ∈ T xn V , • Dφ n (x n )v n → 0, • v n → v ∞ = 0
Let (x n ) n∈N be a sequence of points of V realizing the maximum of the function Scal φ * n gn . The manifold being compact, we can assume that this sequence is convergent to a point x ∞ . We can also assume that the sequence (φ n (x n )) n∈N converges to y ∞ . Of course x ∞ (resp. y ∞ ) realizes the maximum of Scal g ′ ∞ (resp. Scal g∞ ). Let (v n ) n∈N be a sequence given by Fact 3. Reproducing the computation p. 471 of [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF], we see that the scalar curvature of g ′ ∞ is constant along the g ′ ∞ -geodesic starting from x ∞ with speed v ∞ (by symmetry the scalar curvature of g ∞ is constant along a geodesic containing y ∞ ):

Scal g ′ ∞ (exp g ′ ∞ (x ∞ , v ∞ )) -Scal g ′ ∞ (x ∞ ) = lim n→∞ Scal φ * n gn (exp φ * n gn (x n , v n )) -Scal φ * n gn (x n ) = lim n→∞ Scal gn (exp gn (Dφ n (x n , v n )) -Scal gn (φ n (x n )) = Scal g∞ (y ∞ ) -Scal g∞ (y ∞ ) = 0.
Hence g ∞ and g ′ ∞ do not belong to O V . It follows from Theorem 4.2 of [START_REF] Mounoud | Dynamical properties of the space of Lorentzian metrics[END_REF] and Proposition 2 that G ∩ O V is open. As O V is dense we just have to show that G is dense in O V in order to prove Theorem 1. Let g be a metric in O V , as we saw earlier we can perturb it in such a way that the maximum of Scal g is realized by only one point p. This point is now fixed by isometry. We choose now U an open subset of V that do not contain p in its closure but whose closure is contained in some normal neighborhood O of p. According to [1, Theorem 3.1] by Beig et al., we can perturb again g in such a way that there are no local Killing fields on U . We choose the perturbation in order that Scal -1 (M g ) = {p}. The new metric has now a finite isometry group (it is 0-dimensional and compact by Proposition 2 as the metric still lies in O V ). Actually, the proof of Proposition 2 implies also that the set of germs of local isometries fixing p is itself compact. It means that any isometry of a perturbation of g with support not containing O can send a geodesic γ 1 starting from p only on a finite number of geodesics that do not depend on the perturbation. Therefore, it is easy to find a perturbation of the metric along γ 1 (with support away from O) in order to destroy these possibilities. Now, any isometry has to fix pointwise γ 1 (we chose a non symmetric perturbation). Repeating this operation for n = dim V geodesics γ 1 , . . . γ n such that the vectors γ ′ i (0) span T p V , we obtain a perturbation of g such that any of its isometries has to be the identity i.e. the perturbed metric is in G.


H Ghazouani

M Tagina

R Zapata
email: zapata@lirmm.fr

Evolution-Based Vision Algorithm with Fuzzy Fitness Function for Obstacle Detection

Keywords: Parisian evolutionary approach, fuzzy correlation, obstacle detection, vision

Introduction

The proposed algorithm is a fast evolution-based vision technique for real-time obstacle detection [START_REF] Cagnoni | Evolutionary Computer Vision: A Taxonomic Tutorial[END_REF]. Based on the Parisian approach [START_REF] Olague | Parisian evolution with honeybees for three-dimensional reconstruction[END_REF] [3], our algorithm evolves a population of 3D particles which constitutes a three-dimensional representation of the scene. Evolution is controlled by a fuzzy fitness function able to deal with uncertain camera measurements, and uses classical evolutionary operators. The result of the algorithm is a set of 3D particles gathered on the surfaces of obstacles.

Fuzzy correlation

Assuming that the images have been rectified to have disparities only along the Y-coordinate axis, a particle is defined as a 3D point (x, y, z) that projects to the pixel (x p , y p ) in the reference image and the pixel (x p , y p +d) in the right image. If the particle is within an opaque object of the environment, then corresponding pixels in the two images will have similar contrasts and similar neighbouring as shown in Figure 1. Conversely, if the particle is within a heterogeneous surface, similarity between corresponding pixels neighbouring will be low. We propose to calculate a fuzzy correlation cost to express the similarity between two corresponding pixels. We have defined three grey classes which are Black class, Average Class, and White class. Each one of the defined classes has its membership function. Let  black (m),  average (m) and  white (m) be the degrees of membership of the pixel m respectively to the black class, the average class and the white class. We have used Gaussian membership functions centred in 0, 255 and 127.5. Two pixels m 1 and m 2 are strongly correlated if they belong to the same class. This proposition can be expressed using fuzzy logic as follow:

    1 2 1 2
m m FC m m m m m m                        .
Thereafter, we will use the notation: FC(x ,y, z) = FC(m 1 , m 2 ) with m 1 = (x p , y p ) and m 2 = (x p , y p +d) projection of the 3D point (x, y, z). FC(x, y, z) is the fuzzy correlation cost of the projections of the particle (x, y, z) into the reference and the right images.

Genetic conception of the algorithm

We have used a fuzzy fitness function that tolerates uncertainty of measurements. are Sobel gradient norms on left and right projections of the particle [START_REF] Jain | Machine Vision[END_REF]. Individual chromosomes are 3D particle coordinates (x, y, z). Population is randomly initialized within the field of view of the reference camera from a distance d min to the camera. Depths are allocated uniformly by distributing the inverse of z between zero and 1/d min . Selection is elitist and deterministic. Bidimensional sharing penalize particles that project into overcrowd area of the image by reducing their fitness functions by K x N; K is the sharing coefficient and N is the number of particles that projects in a sharing radius R around the current particle. Mutation operator that allow extensive exploration of search space uses an approximation of a Gaussian noise that will be added to (x, y, z) parameters. Z mutation is to add Gaussian noise to the inverse of z so that the variance of z mutation will be coherent with the density of particles that decreases with distance:

min min new z d z d z noise    
.

Thereby, we have used mutations variances  x ,  y , and  z equal to R to be in the same order as the mean distance between two neighbouring particles. Mutation of x and y is calculated in a way to have independence between variance of the displacement of the image of the particle and its depth:

    1 new old focal x x z noise d     
. The noise chosen for x and y is Gaussian and its variance is in the same order as the projection of the distance of a particle to its nearest neighbour . We have defined a barycentric crossover. Given two parents M and N, the algorithm gives their offspring F as:

Figure 1 .

 1 Figure 1. Pixels M L and M R are projections of particle M, are strongly correlated, pixels N L and N R receive light reflections from two objects with different contrasts.

  randomly chosen in [0, 1]). An example of execution in a robot environment scene is illustrated in Figure2.

Figure 2 .

 2 Figure 2. Convergence of our algorithm after 100 generations.