Archive for 4th October 2008

Parallel transport in manifolds with density

When I spoke on Manifolds with Density (see Chapter 18 of the 2008 edition of my Geometric Measure Theory book)  at PIMS at the University of Calgary in September, 2008, Larry Bates asked for a generalization of parallel transport to an n-dimensional manifold M with density e^\psi. For n=2, the generalized curvature \kappa of a curve of Riemannian curvature \kappa_0 involves the log \psi of the density as well:

\kappa = \kappa_0-\frac{d\psi}{dn}. 

Since this describes the rate at which the unit tangent vector is turning, it can be used to define parallel transport.

            For n > 2, the generalized curvature vector \kappa should be the Riemannian curvature vector minus the normal component of gradient of the log of the density. Along with the unit tangent,  \kappa determines a plane. Infinitesimally, vectors normal to the plane are transported by classical parallel transport. For vectors in plane, parallel transport is determined by generalized curvature. 

            Are there any applications?