Related papers: On geometry of frame bundles

Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the…

We provide formulas for Riemannian gradient and Levi-Civita connection for a family of metrics on fixed-rank matrix manifolds, based on nonconstant metrics on Stiefel manifolds.

As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian…

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms,…

Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$…

A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M, whose fibers are the spaces of symmetric, positive-definite bilinear forms on the tangent spaces of M, which represent the Rieman?nian metrics. In this work, we aim…

We explicitly derive the Christoffel symbols in terms of adapted frame fields for the Levi-Civita connection of a Lorentzian $n$-manifold $(M, g)$, equipped with a prescribed optical geometry of K\"ahler-Sasaki type. The formulas found in…

We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of K\"ahlerian Ricci-flat manifolds in four real…

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a…

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce…

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical…

The L\'evi-Civita connection of a Riemannian manifold is a metric (compatible) linear connection, uniquely determined by its vanishing torsion. It is extremal in the sense that it has minimal torsion at each point. We can extend this idea…

In this paper, first, we introduce the Berger-type deformed Sasaki metric on the cotangent bundle $T^{\ast}M$ over a K\"{a}hlerian manifold $(M^{2m}, J, g)$ and investigate the Levi-Civita connection of this metric. Secondly, we present the…

We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant…

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the…

Consider a smooth manifold with a smooth metric which changes bilinear type from Riemann to Lorentz on a hypersurface $\Sigma$ with radical tangent to $\Sigma$. Two natural bilinear symmetric forms appear there, and we use it to analyze the…

Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $% \widetilde{g}_{\nabla,c}$…

We build an analogue for the Levi-Civita connection on Riemannian manifolds for sub-Riemannian manfiolds modeled on the Heisenberg group. We demonstrate some geometric properties of this connection to justify our choice and show that this…

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the…

This paper considers 4-dimensional manifolds upon which there is a Lorentz metric, h, and a symmetric connection and which are originally assumed unrelated. It then derives sufficient conditions on the metric and connection (expressed…