## 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 . For n=2, the generalized curvature of a curve of Riemannian curvature involves the log of the density as well:

=

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 should be the Riemannian curvature vector minus the normal component of gradient of the log of the density. Along with the unit tangent, 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?