Let M be a simply connected complete Kähler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M . Let Uν be the unit normal bundle of N in M . We prove that if a distance function ρ is differentiable at v ∈ Uν , then ρ is also differentiable at −v.

Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...

In this paper, we introduce the new notion of quasi bi-slant submanifolds almost Hermitian manifolds. Necessary and sufficient conditions for integrability distributions which are involved in definition such a Kaehler manifold obtained. We also investigate necessary these manifolds to be totally geodesic study geometry foliations determined by above distributions. Finally, obtain submanifold lo...

A weakly reflective submanifold is a minimal of Riemannian manifold which has certain symmetry at each point. In this paper we introduce notion into class proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist many infinite dimensional PF spaces. particular fiber the parallel transport map shown to be reflective. These imply homogeneous are not totally geodesic, unlike fi...

The object of this paper is to study invariant submanifolds M of Kenmotsu manifolds M̃ admitting a quarter symmetric metric connection and to show that M admits quarter symmetric metric connection. Further it is proved that the second fundamental forms σ and σ with respect to LeviCivita connection and quarter symmetric metric connection coincide. Also it is shown that if the second fundamental f...

this article concerned on the study of signature submanifolds for curves under lie group actions se(2), sa(2) and for surfaces under se(3). signature submanifold is a regular submanifold which its coordinate components are dierential invariants of an associated manifold under lie group action, and therefore signature submanifold is a key for solving equivalence problems.

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro many their generalizations, have only finitely finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for submanifolds dimension at least $2$ are maximal, i.e., not properly contained in a proper submanifold ambient $n...