This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L2-Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the L2-Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the L...

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)metric, whose universal covers Q M satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that @1F Š S ,! S Š @1 Q M are nontrivial knots. As a consequence, we obtain that the group 1.M/ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonposit...

Let M denote a compact real hyperbolic manifold with dimensionm 2: 5 and sectional curvature K = I , and let 1: be an exotic sphere ofdimension m. Given any small number t5 > 0 , we show that there is a finitecovering space M of M satisfying the following properties: the connectedsum M#1: is not diffeomorphic to M, but it is homeomorphic to M; M#1:supports a Riemannian metri...

In computational anatomy, one needs to perform statistics on shapes and transformations, and to transport these statistics from one geometry (e.g. a given subject) to another (e.g. the template). The geometric structure that appeared to be the best suited so far was the Riemannian setting. The statistical Riemannian framework was indeed pretty well developped for finite-dimensional manifolds an...

If π : M → B is a Riemannian Submersion and M has positive sectional curvature, O’Neill’s Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from...

We present a novel geodesic approach to segmentation of white matter tracts from diffusion tensor imaging (DTI). Compared to deterministic and stochastic tractography, geodesic approaches treat the geometry of the brain white matter as a manifold, often using the inverse tensor field as a Riemannian metric. The white matter pathways are then inferred from the resulting geodesics, which have the...

It is proved that in the Voronoi model for percolation in dimension 2 and 3, the crossing probabilities are asymptotically invariant under conformal change of metric. To deene Voronoi percolation on a manifold M , you need a measure , and a Riemannian metric ds. Points are scattered according to a Poisson point process on (M;), with some density. Each cell in the Voronoi tessellation determined...

The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the formK D(H;K) = P i;j ( i; j) TrPiHPjK when P i iPi is the spectral decomposition of the foot point D and the Hermitian matrices H;K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7! G(D) is a ...

The L-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L-metric.

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) nonRiemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler met...