The Length of Harmonic Forms on a Compact Riemannian Manifold

We study $n$ dimensional Riemanniann manifolds with harmonic forms of constant length and first Betti number equal to $n-1$ showing that they are 2-steps nilmanifolds with some special metrics. We also characterise, in terms of properties on the product of harmonic forms, the left invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.Comment: 13 page

A splitting theorem for higher order parallel immersions

We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.Comment: 10 page