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 \(E\longrightarrow M\), 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.
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Notes
These interactions are important, specially for the above problem when we think of the pseudo-Riemannian case. However, even for this situation, for the weighted Sasaki pseudo-Riemannian structures, defined by obvious sign change in (6), we believe the metric completeness of the base manifold is still the sufficient condition, with the same arguments as above.
Indeed, after reading both articles, the present author does not find the significant reason for this attribution and he seems not to be the only; in [20] the choice is referred as a matter of inspiration.
We remark that the structures of generalized Sasaki type with weight functions dependent of the base point \(x\in M\), rather than the squared-radius r, have been studied by the author in [5].
One may say the close relations between metric and complex structures start with the twist \(\varphi _1,\varphi _2\mapsto \psi ,\overline{\psi }\).
Everywhere possible, we omit the ± which is attached to each 2-form and vector bundle.
The \(e^i,\ i=1,2,3\), have norm 2 for the usual metric on 2-forms, but indeed it is \(\frac{1}{2}\) of this that is used in [7]. In particular, the notation here for r refers to the half squared-radius r mentioned there.
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The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA Grant Agreement No. PIEF-GA-2012-332209.
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Albuquerque, R. On vector bundle manifolds with spherically symmetric metrics. Ann Glob Anal Geom 51, 129–154 (2017). https://doi.org/10.1007/s10455-016-9528-y
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DOI: https://doi.org/10.1007/s10455-016-9528-y