Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on $(h,\Phi )$-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic...

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

In a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space), real hypersurfaces admit an almost contact metric structure $(\phi ,\xi ,\eta ,g)$ induced from the ambient space. As a matter of course, many geometers have investigated real hypersurfaces in a nonflat complex space form from the viewpoint of almost contact metric geometry. On the other hand, it is known that the tensor field $h$$...$

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...

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.

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*}M,\overline{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces to the...