Let E be a vector bundle over a compact Riemannian manifold M. We construct a natural metric on the bundle space E and discuss the relationship between the killing vector fields of E and M. Then we give a proof of the Bott-Baum-Cheeger Theorem for vector bundle E.
In this paper we show that the de Rham and the Signature operators on a Riemannian manifold are all isomorphic to some twisted Atiyah-Singer operators. Then the local index theorem and local Lefschetz fixed point formulas of these operators can be obtained from the corresponding theorems of twisted Atiyah-Singer operators.
