 I'm going to talk in English and for first of all I would like to thank the organizers for this opportunity to present my work. I'm going to talk about Voroneuil diagram on Riemannian surface that's a work I'm currently doing as a PhD student with Pierre Calca the University of Rouen and Natalaine Enriquez at the University of Nanterre-Paris West. So what's the main idea? We draw a Voroneuil diagram on a surface and we would like to recover the local geometry of the surface from the statistics of the Voroneuil says. So this requires to show a link between the main characteristics of the Voroneuil says such as area, number of vertices or perimeter and the local characteristics of the surface in what follows it would be the Gaussian curvature. So before going forward some notations in this talk S would be a Riemannian surface with this Riemannian matrix D and its area measure DX induced by the metric and phi will be a nomogenous Poisson point process with respect to this metric and we will be interested by the local geometry around a fixed point X node of S and to do this we will investigate the mean number of vertices of the Voroneuil cells of X node. So the talk is divided in two parts in the first part we investigate for the simplest surface we can find which is the sphere of constant curvature K and I will give a geometric proof to get an exact formula for the mean number of vertices in this case and in a second part I will show how to adapt this method with geometric tools of Riemannian geometry to get an expansion at high intensity for the mean number of vertices on an arbitrary surface. So let's start with the sphere in the case of the sphere of constant curvature K that means it has reduced one of a square root of K. We have this formula so the the mean number of vertices we have a 6 at the beginning which is the Euclidean constant because we if you are in the plane we have 6 for the mean number of vertices and it is corrected by two terms that go to 0 when lambda goes to infinity the first one is of order 1 over lambda and the second one decreases exponentially fast when lambda goes to infinity. So we can note that this expression only depends of the ratio K over lambda this can be explained because by rescaling the sphere we can vary the intensity. So mice in the 70s have proved the same result but for a fixed number of vertices on the of points on the sphere and its proof relies on the correspondence between the vertices of the diagram and the faces of a convex polyhedral and uses a lower formula. So this proof is only available for the sphere and since we have we are interested for more general surface I'm going to introduce a proof a more geometric proof to get this exact formula. So the proof is divided in four steps the first one is since we have to count the vertices we have to characterize it and in fact each vertex is just the circumcentre of x-node and two points of the processes phi and this ball is empty of other points of the process phi contains no other points of the processes phi and this characterize the vertices of C. So to count the vertices of C we just have to sum over the pair of points of phi and since on the sphere three points always define two circumsquite balls we have to sum over the two circumsquite balls. So with this expression we can apply the well-known make a slivak formula to get a double integral and to compute this double integral we are going to do two changes of variable. The first one is the simplest we just use spherical coordinates. So after switching in spherical coordinates we give this expression and we still have to compute the volume of the two circumsquite balls to evaluate this integral. So to do this we make a second change of variables which is a Blaschke-Pekkan-Chim type change of variable which is like this so the idea is to replace the spherical coordinates of x1 and x2 by the spherical coordinates of the circumcenter z and defining the position of x1 and x2 on the boundary of the ball by a theta 1 and theta 2. So thanks to the spherical trigonometric formula we can have these relations on the right and we can compute the Jacobian of the change of variable and so we get this expression where i is integral over all the possible position of x1 and x2 on the boundary and in this new coordinates we have exact formula for the volume of the two circumsquite balls like this and so we can after computing all this get the expression I just announced. So the idea now is to do this for a more general surface so we take an arbitrary surface and we would like to find a way to adapt this method to to get the mean number of vertices but it's quite unlikely that we could get an exact formula in the case of a more general surface so we are I'm going to show how to adapt this method but for to get an asymptotic expansion at high intensity. So the steps will be almost the same we have to characterize the vertices of C and it will not be so different as for the sphere then we have to apply make a Sniak formula and at this moment we will show that only points close to x to x node will contribute significantly in the expansion so we will make a distinction between points near from x near to x node and far from x node and then we will need the two changes of variables the same as before so it will require us to define geodesic polar coordinates to replace the spherical coordinates we used on the sphere and to make a Blaschke-Petkin-Chimpta change of variables the same as before but here in the more general surface we haven't the trigonometric formula to get the expression of these change of variables so we haven't an exact change of variables here and there is an additional step to find the volume of the geodesic volume in the case of the sphere after the second change of variables we had exact formula for the volume of the two circumscribe balls that not the case of in the case of a more general surface so we have to compute this also so let's go step one to characterize vertices of C the same as before we have to sum over the pair of points of phi and to sum over the circumscribe balls which is not unique and we have to apply the maker's linear formula but no for points far from x node we can see that the circumscribe ball will have two large volume and so with the exponential minus lambda in front of this volume we can show that it decreases exponentially fast when lambda goes to infinity and it will be negligible for points around x node we can show it is shown that only one circumscribe ball will contribute the other will be too large and it will contribute negligibly but for the unique circumscribe balls we need similar changes of variable as for the sphere so the first one is to replace the spherical coordinates we used in the case of the sphere and it is replaced by the geodesic polar coordinates geodesic polar coordinates I just are just the polar coordinates of the tojan tangent plane at x node but projected on the surface by the exponential map concretely let's take r and phi if you start from x node and if you go in the direction given by phi for length r you get a point on the on your surface and it has polar coordinates r and phi simply see and since we have to make a change of variable we have to compute the Jacobian associated to this change and in fact we haven't any information about the Jacobian F here but we have a comparison theorem the Roche Roche theorem which is that if the Gaussian curvature is bonded from below and from above by two constants then F is also bonded from below and from above by two functions which are exactly the corresponding functions on the sphere of constant curvature small delta and big delta so since we are in a closed neighborhood around x node and since the Gaussian curvature is continuous we are going to apply this theorem with small delta and big delta close to k of x node so we can make this change of variables and get this expression so the big O here we present all the negligible contributions of the ball I just forgot and for the unique balls which contributes we have make this change of variable an approximate change of variable and now we have we still have to compute the volume of the circumsqy ball and to do this we make the second change of variables which is the Blaschke-Petkin-Cintiq change of variables but here we would like to replace the polar coordinates of x1 and x2 by the geodesic polar coordinates of z and defining the position of x1 and x2 on the boundary of the ball by theta 1 and theta 2 but we have not the relations between all these variables but we have we need and we have another comparison result which is the toponogov theorem toponogov theorem allows us to compare small triangles on surfaces like this it is if the Gaussian curvature is bounded from below and from above by two constants as for a Roche theorem then we can compare small triangles on S with triangles on the spheres of constant curvature small delta and big delta in that way that let's take two segments small segments of fixed lengths with a fixed angles in the between the two segments and if you focus on the distance between the two extremal points then toponogov same that larger the curvature is smaller the distance will be in between the two points this can be explained because when the curvature is large the geodesic tend to get closer faster so the distance between the two points is a smaller so this theorem with the trigonometric formula on the sphere of constant curvature small delta and big delta for small delta and big delta close to k of x0 as before we can get this change of variables so we have an approximation of the relations between the variables and so we can compute an approximate Jacobian which is almost the same as before so we have this expression i is exactly the same integral as for the sphere and in this interval we still have to compute the volume of the circle square ball but this circle square ball in the new coordinates it just a ball of center z and radius r and this volume is given by the Bertrand decay piezo theorem which says that the volume of small geodesic ball of radius r it is just the volume of a nucleon ball of radius r corrected by a term of order r4 depending of the curvature at z but in our case since we are in a closed neighborhood the point z is really close to x0 so if we replace k of z by k of x0 in this expression the error we make is contained in the remainder term a small o of r4 so we can do this and we get this expression there and this integral it remains to compute is a famous form and it is classically computed by Laplace method and we can get this expression for the mean number of vertices so we have this expansion for high intensity so we recover the six which correspond to the Euclidean case and we have a term which goes to zero when lambda goes to infinity which is exactly the same as for the sphere replacing k by k of x0 and with the remaining term after so just to finish before lunch on surface I just talked about it we have shown a link between the mean number of vertices and the Gaussian curvature I presented this result in a positive curvature surface but this result is still available for surveys of negative curvature so it extends also the result of Iso-Kawa in 2000 on the hyperbolic plane I mentioned in the introduction other characteristics such as area and perimeter but I'm not to talk about this now and now we are going on the dimension greater than 3 so we are interested on the local geometry of manifold of dimension greater than 3 and up to some details we have to check but some weeks ago we have proved that there is a link between the mean number of vertices and the scalar curvature in the case for manifold of dimension n and now we are interested by studying other characteristics to get other curvatures because we would like to recover more information about the the local geometry of the surface other than the scalar curvature and there is also a very quite long to-do list in this in this work so I will stop here and thank you for your attention merci en all temps pour quelques questions so do you have an idea of the concentration properties for nfc concentration I order moments in particular are in the last line and it is quite difficult to compute for variance even in the case on in the Euclidean case it is given by some disgusting integral formula and it it requires to compute some volumes of union of balls and it is not really easy to compute it in in this case so I haven't yet this question dans ce cas on remercie encore Aurélie et merci à tous les orateurs de la matinée