Infinitesimal Isometries along Curves and Generalized Jacobi Equations
نویسنده
چکیده
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. The complete prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, it becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is called an infinitesimal isometry along the curve, or a Killing transport. We show that a Killing transport is exactly a rigid variation of the curve, and that Killing transport is parallel translation for a connection on the jet 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 full local existence of infinitesimal isometries.
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