Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions defined...

In the present paper we study naturally reductive homogeneous $(\alpha ,\beta )$-metric spaces. We show that for homogeneous $(\alpha ,\beta )$-metric spaces, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space, given in the literature, are equivalent. Then, we compute the flag curvature of naturally reductive homogeneous $(\alpha ,\beta )$-metric spaces.

In some other context, the question was raised how many nearly Kähler structures exist on the sphere ${𝕊}^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda =12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel ${\mathrm{G}}_2$-structures on the round sphere ${𝕊}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

We introduce a torsion free linear connection on a hypersurface in a Sasakian manifold on which we have defined in natural way a $CR$-structure of $CR$-codimension 2. We study the curvature properties of this connection and we give some interesting examples.

In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space ${ℍ}^{n+1}$, that is, complete hypersurfaces of ${ℍ}^{n+1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R=aH+b$ for some $a$, $b\in ℝ$. In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of ${ℍ}^{n+1}$. Furthermore, a rigidity result...

We prove that there are at least two new non-naturally reductive $\mathrm{Ad}\left(\mathrm{Sp}\right(l)\times \mathrm{Sp}(k)\times \mathrm{Sp}(k)\times \mathrm{Sp}(k\left)\right)$ invariant Einstein metrics on $\mathrm{Sp}(l+3k)$$(k<l)$. It implies that every compact simple Lie group $\mathrm{Sp}\left(n\right)$...

We study the first eigenvalue of the Jacobi operator on closed hypersurfaces with constant mean curvature in non-flat Riemannian space forms. Under an appropriate constraint on the totally umbilical tensor of the hypersurfaces and following Meléndez's ideas in J. Meléndez (2014) we obtain a new sharp upper bound of the first eigenvalue of the Jacobi operator.

We give some new methods to construct nonharmonic biharmonic maps in the unit n-dimensional sphere 𝕊ⁿ.

In this paper we present new examples of $(2n+1)$-dimensional compact cosymplectic manifolds which are not topologically equivalent to the canonical examples, i.e., to the product of the $(2m+1)$-dimensional real torus and the $r$-dimensional complex projective space, with $m,r\ge 0$ and $m+r=n.$ These new examples are compact solvmanifolds and they are constructed as suspensions with fibre the $2n$-dimensional real torus. In the particular case $n=1,$ using the examples obtained, we conclude that a $3$-dimensional compact flat orientable...