SOME RESULTS ON GEODESIC CURVES IN MANIFOLDS WITH DENSITY

Abstract

In this article, we propose some useful notions in manifold with den

sity: 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