arXiv research

The study examines non-negative curvature and conullity of curvature tensors.

Study conullity two manifolds with constant scalar curvature.

Study submanifolds with relative nullity in space forms using splitting tensor.

We find necessary and sufficient conditions for a complete $n$-dimensional Riemannian manifold of finite volume, whose curvature tensor has nullity at least $n-2$, to be a geometric graph manifold. In the process, we show that Nomizu's conjecture, well known to be false in general, is true for manifolds with finite vol…

Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…

The study classifies Riemannian manifolds with curvature nullity.

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $ν$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two tr…

A flow of metrics, $g_t$, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional $S(g)$ is a symmetric $(0,2)$-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. W…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…