On vector bundle manifolds with spherically symmetric metrics

TL;DR

This paper develops a framework for constructing and analyzing weighted spherically symmetric metrics on vector bundle manifolds, exploring their properties, curvature, and applications to special holonomy G2 manifolds.

Contribution

It introduces a generalized class of spherically symmetric metrics on vector bundles with fiber-dependent weights, and studies their geometric properties and holonomy.

Findings

Defined a new class of weighted spherically symmetric metrics

Analyzed curvature equations and properties of these metrics

Computed holonomy for Bryant-Salamon type G2 manifolds

Abstract

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 metric connection. The tangent bundle of $E$ admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of $E$; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type $\mathrm{G}_2$ manifolds.