Codazzi Tensor Fields, Curvature and Pontryagin Forms

for arbitrary vector fields X, Y, Z. In this case, the self-adjoint section B of End TM, characterized by g(BX, Y) = b(X, Y), will also be called a Codazzi tensor. The Codazzi tensor b will be called non-trivial if it is not a constant multiple of the metric. The aim of the present paper is to study some geometric and topological consequences of the existence of a non-trivial Codazzi tensor on a given Riemannian manifold. Results of this type were obtained by Bourguignon [3], who proved that the existence of such a tensor imposes strong restrictions on the curvature operator [3, Theoreme 5.1 and Corollaire 5.3] and, as a consequence, obtained the following theorem [3, Corollaire 7.3]: a compact orientable Riemannian four-manifold admitting a non-trivial Codazzi tensor with constant trace must have signature zero. Our main results consist in generalizing these theorems, in particular in seeing what can be said when the assumption on the trace is dropped. In § 2 of this paper we observe that, in the C°° category, every manifold admits a Riemannian metric with a non-trivial Codazzi tensor (Example 7), so that topological consequences may be expected only if some sort of analytic behaviour is assumed. Section 3 is devoted to the particular consequences of the existence of a non-trivial Codazzi tensor B for the structure of the curvature operator (Theorem \):for any point x of the manifold M and arbitrary eigenspaces Vx, V^ ofBx, the span Vx A V^cz A TXM of all exterior products of elements of Vx and V^ is invariant under the curvature operator Rx acting on 2-forms. As a consequence, we obtain in §4 a relation between the eigenspaces of any Codazzi tensor and the Pontryagin forms (Propositions 3 and 4), which, together with an extra argument for the case of a Codazzi tensor having only two distinct eigenvalues (Lemma 1), implies that a compact orientable Riemannian four-manifold (M, g) admitting a non-trivial Codazzi tensor b must have signature zero unless the restriction ofb to some non-empty open subset ofM is a constant multiple ofg (Theorem 2). Another consequence of Proposition 4 is that for any n-dimensional Riemannian manifold with a Codazzi tensor having n distinct eigenvalues almost everywhere, all the real Pontryagin classes are zero (Corollary 3).