We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian $g$-natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.

We will prove that if an open subset of $ℂP^n$ is isometrically immersed into $ℂP^m$, with $m<(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $ℍP^n$ isometrically immersed into $ℍP^m$, with $m<(4/3)n-5/6$, then the image is totally geodesic.

In this paper we survey some recent results on rank one symmetric space.

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class...

Let $(M,g)$ be a Riemannian manifold, $L\left(M\right)$ its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi–Civita connection and curvatures of these metrics.

We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.