Killing Vector Fields on Complete Riemannian Manifolds

Abstract

We discuss Killing vector fields with finite global norms on complete Riemannian manifolds whose Ricci curvatures are nonpositive or negative.1.It is well known that if a compact Riemannian manifold has nonpositive Ricci curvature then every Killing vector field is a parallel vector field (cf.[3]).In this note, we discuss Killing vector fields with finite global norms on complete Riemannian manifolds.One of our results is that if Af is a complete Riemannian manifold with nonpositive Ricci curvature then every Killing vector field on M with finite global norm is a parallel vector field.This is a generalization of the above well-known result.We also discuss the volume of a complete noncompact Riemannian manifold with nonpositive Ricci curvature.Our ideas are based on those of the papers of A. Andreotti and E. Vesentini [1] and, especially, H. Kitahara [2].We shall be in the C "-category.The manifolds considered are connected and orientable.The indices n, i, j, k, . . .run over the range (1, 2, . . ., n} and the Einstein summation convention will be used.2. Let M he an n-dimensional complete Riemannian manifold and g (resp.V) its Riemannian metric tensor field (resp.its Levi-Civita connection).Let {U: (xx, . . ., x")} denote a local coordinate system on M. gtj denotes the components of g and (g'J) denotes the inverse matrix of the matrix (gu).We set V, = Va/dx¡ and V = g%.Let A*(Af) be the space of all i-forms on M and Ac,(A/) the subspace of AS(M) composed of forms with compact supports, tj G AS(M) may be expressed locally as r, = (l/s\)r1i¡...¡idxi'A-■ ■ Adx': Let < , > denote the local scalar product; the global scalar product « , >> is defined by «&*»-/<£?>•!-f IA*t?J M J M for any £, r/ G A^Af), where * denotes the star operator (cf.[4]).Let L2(M) he the completion of A¿(Af) with respect to the scalar product « , >>. d: A*(M)-> AS+X(M) denotes the exterior derivative and 5: A^Af)-* Ai_1(Af) is defined by