2009/06/04 by Robert L. Foote, Foote, Robert L., Chong-Kyu Han +4
Mathematics · Medicine · Physics and Astronomy · #35N10 #53B21 (primary) #53C29 (secondary) #58J60 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Ophthalmology and Eye Disorders #math.AP #math.DG #msc:35N10 #msc:53B21 #msc:53C29 #msc:58J60
paper · pdf · doi:10.48550/arxiv.0906.0821
24 pages. See also http://persweb.wabash.edu/facstaff/footer/Abstracts.HTM. Minor additions/corrections. To appear in J. Geometric Analysis
openalex publication_date 2009/06/04 · arxiv created 2011/09/15 · arxiv updated 2011/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.