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The length of harmonic forms on a compact Riemannian manifold

Authors: Paul-Andi Nagy and Constantin Vernicos
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2501-2513
MSC (2000): Primary 53C20, 58J50
Published electronically: January 23, 2004
MathSciNet review: 2048527
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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.

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Additional Information

Paul-Andi Nagy
Affiliation: Université de Neuchâtel, Institut de Mathématiques, Rue Emile Argand 11, CH-2007 Neuchâtel, Switzerland
Address at time of publication: Humboldt-Universität zu Berlin, Institut für Mathematik, Sitz: Rudower Chaussee 25, D-10099 Berlin, Germany

Constantin Vernicos
Affiliation: Université de Neuchâtel, Institut de Mathématiques, Rue Emile Argand 11, CH-2007 Neuchâtel, Switzerland

Keywords: Length of harmonic forms, isosystolic inequalities, spectrum of the Laplacian
Received by editor(s): January 31, 2003
Received by editor(s) in revised form: April 30, 2003
Published electronically: January 23, 2004
Additional Notes: The first author was partially supported by european project ACR OFES number 00.0349
The second author was partially supported by european project ACR OFES number 00.0349 and a grant of the FNRS 20-65060.01
Article copyright: © Copyright 2004 American Mathematical Society