The Lichnerowicz–Obata Theorem on Sub-Riemannian Manifolds with Transverse Symmetries

Riemannian Symmetries in Flag Manifolds

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of Z2 -symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the Z2-symmetric structure to be naturally reductive are detailed for the flag manifold SO(5)/SO(2)× SO(2)× SO(1).

On Sublaplacians of Sub-riemannian Manifolds

In this note we address a notion of sublaplacians of sub-Riemannian manifolds. In particular for fat sub-Riemannian manifolds we answered the sublaplacian question proposed by R. Montgomery in [21, p.142]

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal m...

The Geometry of Sub-riemannian Three-manifolds

The local equivalence problem for sub-Riemannian structures on threemanifolds is solved. In the course of the solution, it is shown how to attach a canonical Riemannian metric and connection to the given sub-Riemannian metric and it is shown how all of the differential invariants of the sub-Riemannian structure can be calculated. The relation between the completeness of the sub-Riemannian metri...

The Hodge theorem for Riemannian manifolds