First and second variation of arc length and energy in Riemannian manifold with density

Abstract

In this paper, we give a definitions of ?-arc length L ? and ?-energy E? of a curve in Riemannian manifold M with density e? and we prove that L2? ? 2(b - a)E?. Also we study the first and second variation of arc length and energy in Riemannian manifold M with density ? = e?.