Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

Irreducibility of the Laplacian eigenspaces of some homogeneous spaces Math. Z. https://doi.org/10.1007/s00209-018-2088-z Mathematische Zeitschrift Irreducibility of the Laplacian eigenspaces of some homogeneous spaces 1 1 David Petrecca · Markus Röser Received: 4 October 2017 / Accepted: 23 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract For a compact homogeneous space G/K , we study the problem of existence of G- invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of G. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, we provide existence results for such metrics on certain isotropy reducible spaces. Keywords Homogeneous spaces · Symmetric spaces · Laplacian · spherical representations Mathematics Subject Classification Primary 53C30; Secondary 58J50 · 22E46 Contents Introduction .................................................. 1 The eigenspaces of the Laplacian of a homogeneous space ........................ 2 Compact symmetric spaces ........................................ 3 Towards the case of non-symmetric homogeneous spaces ........................ References ................................................... Introduction Let (M, g) be a compact Riemannian manifold and let Δ be the Laplace operator of g acting on smooth functions. It is an elliptic, self-adjoint and non-negative operator, so its non-zero eigenvalues λ are positive and the corresponding eigenspaces E are finite-dimensional. i http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Irreducibility of the Laplacian eigenspaces of some homogeneous spaces

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-018-2088-z
Publisher site
See Article on Publisher Site

Abstract

Math. Z. https://doi.org/10.1007/s00209-018-2088-z Mathematische Zeitschrift Irreducibility of the Laplacian eigenspaces of some homogeneous spaces 1 1 David Petrecca · Markus Röser Received: 4 October 2017 / Accepted: 23 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract For a compact homogeneous space G/K , we study the problem of existence of G- invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of G. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, we provide existence results for such metrics on certain isotropy reducible spaces. Keywords Homogeneous spaces · Symmetric spaces · Laplacian · spherical representations Mathematics Subject Classification Primary 53C30; Secondary 58J50 · 22E46 Contents Introduction .................................................. 1 The eigenspaces of the Laplacian of a homogeneous space ........................ 2 Compact symmetric spaces ........................................ 3 Towards the case of non-symmetric homogeneous spaces ........................ References ................................................... Introduction Let (M, g) be a compact Riemannian manifold and let Δ be the Laplace operator of g acting on smooth functions. It is an elliptic, self-adjoint and non-negative operator, so its non-zero eigenvalues λ are positive and the corresponding eigenspaces E are finite-dimensional. i

Journal

Mathematische ZeitschriftSpringer Journals

Published: May 31, 2018

References

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