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 −→ 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 G 2 manifolds.

Self-duality and associated parallel or cocalibrated G_2 structures

“…But a central geodesic remains in a fixed radius, hence it cannot be extended indefinitely. The holonomy equal to G 2 now follows by the results in [Alb14].…”

“…We note the incompleteness of the metric is in great contrast with the elliptic geometry case. The end of the proof is accomplished with a general method found in [Alb14]. Which also gives a new proof of the s > 0 case, i.e.…”

Notes on the Sasaki metric

“…The topology of the tangent bundle does not prevent the geodesics from not being defined for all t ∈ R. In other words, we are sincerely convinced (T M, g) is a complete Riemannian manifold, as long as M is complete -cf. discussion and similar metrics referred in [5].…”

“…dual to π * ∂ i , π ⋆ ∂ i , cf. (5), show the extension is independent of the torsion-free connection and choice of chart. π * α is the usual pull-back.…”

Conformal vector fields on vector bundle manifolds with spherically symmetric metrics

“…As Riemannian manifolds, vector bundles have been considered by many authors ( [3], [6], [12]). An interesting step has been taken by R. Albuquerque, who introduced the important class of spherically symmetric metrics on vector bundle manifolds.…”