| Abstract: | Given a closed connected Riemannian manifold M and a connected Riemannian manifold N, we consider fiberwise, i.e. M×{z}, z ? N{M\times \{z\}, z\in N}, volume non-increasing diffeomorphisms on the product M × N. Our main theorems show that in the presence of a certain cohomological condition on M and N such diffeomorphisms must map a fiber diffeomorphically onto another fiber and are therefore fiberwise volume preserving.
As a first corollary, we show that the isometries of M × N split. As a second corollary, we prove a special case of an extension of Talelli’s conjecture. |
