First and Second Variation of Arc Length and Energy in Riemannian Manifold with Density

摘要

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