## Tohoku Mathematical Journal 2000 June SECOND SERIES VOL. 52, NO. 2

 Tohoku Math. J. 52 (2000), 235-249

Title ON THE SECOND VARIATION OF THE IDENTITY MAP OF A PRODUCT MANIFOLD

Author Ali Fardoun and Andrea Ratto

(Received September 22, 1998, revised May 6, 1999)
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.

1991 Mathematics Subject Classification. Primary 58E20; Secondary 53C21, 58G25.

Key words and phrases. Harmonic maps, Second variation, Jacobi fields, Spectrum of the Hodge-Laplacian.