naa a22 4500 510810365 CHVBK 20180411083439.0 cr unu---uuuuu 180411e20130101xx s 000 0 eng 10.1007/s12220-011-9250-8 doi (NATIONALLICENCE)springer-10.1007/s12220-011-9250-8 Infinitesimal Isometries Along Curves and Generalized Jacobi Equations [Elektronische Daten] [R. Foote, C. Han, J. Oh] On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. Acompletely determined prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, this becomes an ODE that generalizes the Jacobi equation. Asolution of this ODE is called an infinitesimal isometry along the curve, which we show to be an infinitesimal rigid variation of the curve. We define Killing transport to be the associated linear isometry between fibers of the bundle along the curve, and show that it is parallel translation for a connection on the bundle related to the Riemannian connection. Restricting to dimension two, we study the holonomy of this connection, prove the Gauss-Bonnet theorem by means of Killing transport, and determine the criteria for local existence of infinitesimal isometries. Mathematica Josephina, Inc., 2011 Infinitesimal isometry nationallicence Killing field nationallicence Completely determined prolongation nationallicence Solvability nationallicence Jacobi equation nationallicence Killing transport nationallicence Holonomy nationallicence Gauss-Bonnet theorem nationallicence Foote R. Department of Mathematics and Computer Science, Wabash College, 47933, Crawfordsville, IN, USA aut Han C. Department of Mathematics, Seoul National University, 151-742, Seoul, South Korea aut Oh J. Department of Mathematics, Seoul National University, 151-742, Seoul, South Korea aut Journal of Geometric Analysis Springer-Verlag 23/1(2013-01-01), 377-394 1050-6926 23:1<377 2013 23 12220 https://doi.org/10.1007/s12220-011-9250-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s12220-011-9250-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Foote R. Department of Mathematics and Computer Science, Wabash College, 47933, Crawfordsville, IN, USA aut NATIONALLICENCE 700 1- Han C. Department of Mathematics, Seoul National University, 151-742, Seoul, South Korea aut NATIONALLICENCE 700 1- Oh J. Department of Mathematics, Seoul National University, 151-742, Seoul, South Korea aut NATIONALLICENCE 773 0- Journal of Geometric Analysis Springer-Verlag 23/1(2013-01-01), 377-394 1050-6926 23:1<377 2013 23 12220 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer