On vector bundle manifolds with spherically symmetric metrics

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 $E\to M$, 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.

Final version

References in corpus (2)

• Homotheties and topology of tangent sphere bundles
• Weighted metrics on tangent sphere bundles

Cited by in corpus (4)

• Notes on the Sasaki metric
• The ciconia metric on the tangent bundle of an almost-Hermitian manifold
• The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids
• On biharmonic functions on vector bundles