We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Introduction Consider a distribution on a manifold M, i.e. subbundle of the tangent bundle Π ⊂ TM . Non-holonomic Riemannian metric is a Riemannian metric g ∈ SΠ on this bundle. We call the pair (Π, g) sub-Riemannian structure. A curve γ : [0, 1] → M is called horizontal if γ̇ is a sect...

we study umbilic (space-like) submanifolds of pseudo-riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo euclidean space and relate this notion to umbilicity. finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.a pseudo-riemannian submanifold m in (a ps...

in this paper, the matsumoto metric with special ricci tensor has been investigated. it is proved that, if  is ofpositive (negative) sectional curvature and f is of  -parallel ricci curvature with constant killing 1-form ,then (m,f) is a riemannian einstein space. in fact, we generalize the riemannian result established by akbar-zadeh.

In an earlier paper we developed the classification of weakly symmetric pseudo–Riemannian manifolds $G/H$, where $G$ is a semisimple Lie group and $H$ reductive subgroup. We derived from cases compact. As consequence obtained Lorentz signature $(n-1,1)$ trans-Lorentzian $(n-2,2)$. Here work out pseudo-Riemannian nilmanifolds $G/H$ for case $G=N \rtimes H$ with compact $N$ nilpotent. It turns th...

A basic question in geometry is to understand compact locally homogeneous manifolds, i.e., those compact manifolds that can be locally modelled on a homogeneous space J\H of a finite-dimensional Lie group H. This means that there is an atlas on a manifold M consisting of local diffeomorphisms with open sets in J\H where the transition functions between these open sets are given by translations ...