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The geometry of k -harmonic manifolds

The geometry of k -harmonic manifolds Abstract An n -dimensional Riemannian manifold is called k -harmonic for some integer k , 1 ≤ k ≤ n - 1, if the k -th elementary symmetric functions of the principal curvatures of small geodesic spheres are radial functions. We prove that k -harmonic manifolds are necessarily 2-stein and show that locally symmetric manifolds which are k -harmonic for one k , are k -harmonic for all k . We then establish some results relating the harmonic and k -harmonic conditions for the class of non-compact harmonic non-symmetric spaces constructed by Damek and Ricci. We also discuss other notions of k -harmonicity and the problem of their equivalence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

The geometry of k -harmonic manifolds

Nicolodi, Lorenzo; Vanhecke, Lieven
Advances in Geometry , Volume 6 (1) – Jan 26, 2006
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18 pages

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Publisher
de Gruyter
Copyright
Copyright © 2006 by the
ISSN
1615-715X
eISSN
1615-7168
DOI
10.1515/ADVGEOM.2006.004
Publisher site
See Article on Publisher Site

Abstract

Abstract An n -dimensional Riemannian manifold is called k -harmonic for some integer k , 1 ≤ k ≤ n - 1, if the k -th elementary symmetric functions of the principal curvatures of small geodesic spheres are radial functions. We prove that k -harmonic manifolds are necessarily 2-stein and show that locally symmetric manifolds which are k -harmonic for one k , are k -harmonic for all k . We then establish some results relating the harmonic and k -harmonic conditions for the class of non-compact harmonic non-symmetric spaces constructed by Damek and Ricci. We also discuss other notions of k -harmonicity and the problem of their equivalence.

Journal

Advances in Geometry – de Gruyter

Published: Jan 26, 2006

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