In the class of real hypersurfaces $M^{2n-1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_n\left(c\right)$ of constant holomorphic sectional curvature $c$$(\ne 0)$ which is either a complex projective space $ℂP^n\left(c\right)$ or...

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection ${\pi }_E$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of ${\pi }_E$.

In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold $(M,g)$ onto a complete Riemannian manifold $(\widehat{M},\widehat{g})$ in terms of developing geodesic triangles of $M$ onto $\widehat{M}$. More precisely, we show that if $A_0:T{{|}_{x_0}M\to T|}_{{\widehat{x}}_0}\widehat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $\gamma$, $\omega$ of $M$ based at $x_0$ the development through $A_0$ of the composite path $\gamma ·\omega$ onto $\widehat{M}$ results in a closed path based at ${\widehat{x}}_0$, then there exists a Riemannian covering map...

Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $\left(K_{x,y}\circ K_{z,u}\right)\left(u\right)=\left(K_{z,u}\circ K_{x,y}\right)\left(u\right)$, where $K_{x,y}\left(u\right)=R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.