Abstract: We give new conditions on positivity of certain
linear combinations of eigenvalues of the curvature operator of a
Riemannian manifold which imply the vanishing of the indices of Dirac
operators twisted with bundles of tensors. The vanishing indices in
turn have topological implications in terms of the Pontryagin classes,
rational cobordism type, and Witten genus of the manifolds. To prove
our results we generalize new methods developed by Petersen and Wink
to apply the Bochner technique to Laplacians on bundles of tensors.