Periodic Solutions with Prescribed Energy on Non-complete Riemannian Manifolds*

Let M* be a C3 finite dimensional manifold and M:M* a connected open subset such that M, ? , ?:R.is a Riemannian manifold with topological boundary ­ M and let V: MªR be a C1 potential function. In this paper we look for curves in M which are periodic solutions of Dt˙x t. sy=RV x t.., 1.1. where Dt˙x t. denotes the covariant derivative of˙x t. along the direction of ˙x t. and =R the Riemannian gradient. The research of periodic solutions of 1.1.of prescribed period has been dealt with in many papers see references in w1x and also w4, 12, 15x.. As 1.1.is an autonomous system, the energy Es12 ˙x,˙x:RqV x. is constant along the solutions of 1.1.and thus it is also interesting to look for periodic solutions of 1.1.with prescribed energy.