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arXiv · 1911.09031

Cones and Cartan geometry

Antonio J. Di Scala·Carlos E. Olmos·Francisco Vittone
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Abstract

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hat\omega$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.

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Antonio J. Di Scala, Carlos E. Olmos, Francisco Vittone. 2019-11-20. Cones and Cartan geometry. https://arxiv.org/abs/1911.09031

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