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A Symmetric Product for Vector Fields and its Geometric Meaning

Lewis, Andrew D. (1996) A Symmetric Product for Vector Fields and its Geometric Meaning. California Institute of Technology , Pasadena, CA. (Unpublished)

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We introduce the notion of geodesic invariance for distributions on manifolds with a linear connection. This is a natural weakening of the concept of a totally geodesic foliation to allow distributions which are not necessarily integrable. To test a distribution for geodesic invariance, we introduce a symmetric, vector field valued product on the set of vector fields on a manifold with a linear connection. This product serves the same purpose for geodesically invariant distributions as the Lie bracket serves for integrable distributions. The relationship of this product with connections in the bundle of linear frames is also discussed. As an application, we investigate geodesically invariant distributions associated with a left-invariant affine connection on a Lie group.

Item Type:Report or Paper (Technical Report)
Additional Information:Submitted to Mathematische Zeitschrift. The author would like to thank Francesco Bullo for many helpful discussions concerning the material in Section 5. This research was financially supported in part by NSF Grant CMS-9502224.
Group:Control and Dynamical Systems Technical Reports
Subject Keywords:symmetric product, linear connections, affine connections, Riemannian geometry, Lie groups
Record Number:CaltechCDSTR:1996.003
Persistent URL:
Usage Policy:You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
ID Code:28112
Deposited By: Imported from CaltechCDSTR
Deposited On:20 Oct 2006
Last Modified:03 Oct 2019 03:29

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