On the second variation of the identity map of a product manifold

Abstract

The main aim of this paper is to compute the index and the nullity of the identity map of $S^n\times S^m$ and $S^n\times T^m$. In order to obtain this we establish a rather general result on the spectrum of the Hodge-Laplacian on $k$-forms on a product manifold, which could prove useful in other contexts.