Conformal and Killing Vector Fields on Complete Non-compact Riemannian Manifolds

Abstract

In this note, we introduce the notion of vector fields with finite global norms, in order to discuss the vector fields on non-compact Riemannian manifolds.It should seem to be natural notion because we have some generalizations of well-known results for compact Riemannian manifolds (cf.[3], [9]).These generalizations are our main results.Our discussions are restricted to conformal and Killing vector fields.We show some examples in which the relations between the volumes of complete non-compact Riemannian manifolds and the global norms of Killing vector fields are discussed.For Killing vector fields with finite global norms, the case of complete non-compact Riemannian manifolds without boundary has stated in [11], and the case of non-compact Riemannian manifolds with boundary has stated in [12].Our idea is based on in [1],[4], [6] and [10].The case of affine and projective vector fields with finite global norms may be discussed similarly, but this case is not stated in this note (cf.[13]).The discussions of different point of views appeared in [5] and [7].We shall be in Coo-category.The manifolds considered are connected and orientable.1. Let M be a complete non-compact Riemannian manifold (without boundary) of dimension m.We denote the Riemannian metric (resp.the Levi-Civita connection) on M by g (resp.17).Let gij denote the components of g with respect to a local coordinate system (xl, ... , x m ), and (gij) denotes the inverse matrix of the matrix (gij).We set l7i=17a/axi and l7i=giipj.For two (0, s)-tensor fields T and S on M, we denote the local scalar product (resp.the global scalar product) of T and S by < T, S) (resp.((T, S»), that is,