Infinitesimal isometries along curves and generalized Jacobi equations

Details Infinitesimal isometries along curves and generalized Jacobi equations Infinitesimal isometries along curves and generalized Jacobi equations

2009-06-04

arxiv.org/abs/0906.0821v3

Mathematics

Differential Geometry

24 pages. See also http://persweb.wabash.edu/facstaff/footer/Abstracts.HTM. Minor additions/corrections. To appear in J. Geo

Scientific paper

A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete prolongation of the Killing equation and then restrict to the curve, and 2) compute the variational equations of a rigid motion of the curve. This results in Killing transport along the curve, which is parallel transport for a related connection on the jet bundle J(TM). We study the curvature and holonomy of this connection. In particular, in dimension two the curvature is the local obstruction to infinitesimal isometries on M.

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