Extending tensors on polar manifolds

Ricardo A. E. Mendes

Extending tensors on polar manifolds

Differential Geometry 2013-08-13 v1

Authors: Ricardo A. E. Mendes

Abstract

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$ , with section $\Sigma\subset M$ and generalized Weyl group $W$ . We show that restriction to $\Sigma$ is a surjective map from the set of smooth $G$ -invariant tensors on $M$ onto the set of smooth $W$ -invariant tensors on $\Sigma$ . Moreover, we show that every smooth $W$ -invariant Riemannian metric on $\Sigma$ can be extended to a smooth $G$ -invariant Riemannian metric on $M$ with respect to which the $G$ -action remains polar with the same section $\Sigma$ .

Keywords

riemannian manifold polar action invariant tensors weyl group metric extension lie group restriction map

Cite

@article{arxiv.1308.2412,
  title  = {Extending tensors on polar manifolds},
  author = {Ricardo A. E. Mendes},
  journal= {arXiv preprint arXiv:1308.2412},
  year   = {2013}
}

Comments

arXiv admin note: text overlap with arXiv:1205.4765

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