The length of harmonic forms on a compact Riemannian manifold

Abstract

Abstract. We study (n + 1)-dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to n showing that they are 2-step nilmanifolds with some special metrics. We also characterize, 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.

On length and product of harmonic forms in Kähler geometry

“…Therefore, in the special case when b 1 (N) = n − 1 it follows that the fibers of the Albanese map must be totally geodesic. Using Chern-Weil theory and an argument that reproduces in part that in section 6 of [3], we showed in [14] that the only possible topologies of manifolds N n which admit a metric satisfying (CL 1 ) and have b 1 (N) = n − 1 are those of 2-step nilmanifolds with 1-dimensional kernel. Equivalently, the above class of manifolds is parametrized by couples (T, ω) where T is a flat (n − 1)-torus and ω is a non zero, integral cohomology class on T .…”

“…Higher dimensional examples, endowed with non-homogeneous Riemannian metrics can be obtained by taking the product with S 1 of the class of nilmanifolds studied in [14].…”

“…In higher dimensions, there are very few general facts known about geometrically formal manifolds; for instance formal metrics satisfy hypothesis (CL p ) for all 1 ≤ p ≤ n − 1 [11]. Note that, by contrast, the class of (non necessarily invariant) metrics on nilmanifolds studied in [14] satisfy hypothesis (CL p ) whenever 1 ≤ p ≤ n − 1 but none of the p-formality hypothesis. Moreover, it is known that certain classes of homogeneous spaces fail to be geometrically formal for cohomological reasons [12].…”

Four-manifold systoles and surjectivity of period map

“…Nontrivial cup product relations lead to stable systolic inequalities [BanK03] (cf. inequality (4.3) below), some of them sharp [BanK2,NV03]. Meanwhile, nontrivial Massey products also admit systolic repercussions, cf.…”

Systems of symplectic forms on four-manifolds

“…In fact, if the induced metric were even only complete, X would become a global tri-holomorphic Killing vector field, in contradiction to [12,Theorem 1 (iii)]. Secondly, Theorem 1.1 can be considered as a local 4-dimensional analogue, for two-forms, of the following result: Theorem [23]. Let (M, g) be a compact Riemannian n-manifold with b 1 (M) = n 1 and such that every harmonic 1-form has constant length.…”