SOME RESULTS ON GEODESIC CURVES IN MANIFOLDS WITH DENSITY

Abstract

In this article, we propose some useful notions in manifold with density: weighted Levi-Civita connection, weighted covariant derivative, weighted geodesic curve, totally weighted geodesic submanifold, weighted energy functional and minimal weighted geodesic curves. We proved that a constant speed curve is a weighted geodesic curve if and only if it is a critical point of the weighted length functional, a curve γ minimizes the weighted energy functional if and only if γ is a minimal weighted geodesic with constant weighted velocity.