The Length of Harmonic Forms on a Compact Riemannian Manifold

Paul-Andi Nagy (1) , Constantin Vernicos (2)

1 Institut für Mathematik [Berlin]
2 IMAG - Institut Montpelliérain Alexander Grothendieck

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.

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Paul-Andi Nagy, Constantin Vernicos. The Length of Harmonic Forms on a Compact Riemannian Manifold. Transactions of the American Mathematical Society, 2004, 356 (6), pp.2501--2513 (electronic). ⟨10.1090/S0002-9947-04-03546-9⟩. ⟨hal-01160279⟩