Infinitesimal Isometries on Compact Manifolds

Let X denote a non-vanishing infinitesimal isometry on a compact Riemannian manifold Mn. Let denote the deRham complex of M. We write i(X) for the operator of interior product, and L(X) the Lie derivative on the elements of A(M). We define E(M) = {u ∈ A(M)| i(X)u = 0, L(X)u= 0}.

Fixed Points of Isometries

The purpose of this paper is to prove the followingTheorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)