Curvature operators and characteristic classes

Abstract:

Given tensors $A$ and $B$ of type $(k, k)$ on a Riemannian manifold $M$ we construct in a natural way a $2k$ form ${F_k}(A, B)$. If $A$ and $B$ satisfy the generalized Codazzi equations then this $2k$ form is closed. In particular if ${R_{2k}}$ denotes the $2k$th curvature operator then ${F_{2k}}({R_{2k, }}{R_{2k}})$ is (up to a constant multiple) the $k$th Pontrjagin class of $M$. By means of a theorem of Gilkey we give conditions sufficient to guarantee that a form constructed from more complicated expressions involving the curvature operators does in fact belong to the Pontrjagin algebra. As a corollary we obtain Thorpe’s vanishing theorem for manifolds with constant $2p$th sectional curvature. If at each point in $M$ the tangent space contains a subspace of a particular type (similar to curvature nullity) we show that certain Pontrjagin classes must vanish. We generalize the result that submanifolds of Euclidean space with flat normal bundle have a trivial Pontrjagin algebra. The curvature operator, ${R_2}$, is interesting in that the components of ${R_2}$ with respect to any orthonormal frame are given by certain universal (independent of frame) homogeneous linear polynomials in the components of the curvature tensor. We characterize all such operators and using this characterization derive in a natural way the Weyl component of ${R_2}$.