Local Invariants of an Embedded Riemannian Manifold

Abstract

Let M be a Riemannian manifold of dimension m which is isometrically embedded in a Riemannian manifold N of dimension me + r. Let G denote the metric on N and GM denote the metric on-M. In this paper, we will study the local invariants of such an embedding. The invariants will be p-form valued polynomials in the derivatives of the metric defined on M. We have studied the case r = 0 in [4] for pseudo-Riemannian manifolds M. The results of this paper can be generalized to the case in which both M and N are pseudo-Riemannian manifolds. We will be considering invariance relative to the following four groups: let v = v, x ,29 Set v, = SO(m) if M is oriented and 0(m) otherwise, ,2 = SO(r) if the normal bundle is oriented and 0(r) otherwise. Let P , denote the space of p-form valued >-invariant polynomials in the derivatives of the metric; we will give a precise definition of these spaces in the first section. Since exterior differentiation is natural, d induces a map d: 9P CP +J. If p < n, the kernel of d modulo the image of d is spanned by the characteristic classes. If p = m, every form is closed. We will give an integral condition which characterizes those invariants which are characteristic forms + dQ for some Q e gJ m-l In Corollary 1.3, we apply these theorems to settle a conjecture of I. M. Singer regarding the Euler class (private communication).