A C∞ function f on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (f ◦σ)′′ ≥ 0 for every geodesic inM . We shall apply this notion in a variety of ways to the study of manifolds of negative or nonpositive curvature. Convexity has, of course, long been associated with negative curvature, but convex function seem...

Let M^2n be a hoph hypersurfaces with parallel ricci operator and tangent to structure vector field in Sasakian space form. First, we show that structures and properties of hypersurfaces and hoph hypersurfaces in Sasakian space form. Then we study the structure of hypersurfaces and hoph hypersurfaces with a parallel ricci tensor structure and show that there are two cases. In the first case, th...

In this paper, we introduce and examine invariant semi-invariant lightlike submanifolds of a poly-Norden semi-Riemannian manifold. Also, obtain some examples such types study the conditions for both integrability totally geodesic foliation description distributions.

A Riemannian manifold satis es the axiom of 2-planes if at each point, there are su ciently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several de nitions of axiom of planes arise naturally when we consider such families of tangent planes...