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The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces

The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces Arch. Math, Vol. 59, 65-79 (1992) 0003-889X/92/5901-0065 $ 4.50/0 9 1992 Birkh/iuser Verlag, Basel The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces By CHRISTIAN BAR O. Introduction. The spin structures of an oriented Riemannian homogeneous space M = G/H can be characterized by lifts of the isotropy representation. These basics are studied in the first section. In the second section we calculate the Dirac operator in algebraic terms, generalizing a well-known formula for symmetric spaces. Using Frobe- nius reciprocity, stated in Section 3, we calculate the Dirac spectrum of S 3 and quotients equipped with Berger metrics. Under the collapse onto S 2 some eigenvalues of S 3 con- verge to those of S 2 while the rest tends to + oe. The real projective space NP 3 equipped with the first spin structure shows the same behaviour whereas all eigenvalues of IIP 3 with the second spin structure tend to + oo. The Berger spheres also show that the dimension of the space of harmonic spinors highly depends on the Riemannian metric, an observation due to Hitchin, see [4]. This paper is part of the author's thesis, see [1]. 1. Spin structures on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archiv der Mathematik Springer Journals

The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces

Archiv der Mathematik , Volume 59 (1) – Feb 4, 2005

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References (3)

Publisher
Springer Journals
Copyright
Copyright © 1992 by Birkhäuser Verlag
Subject
Mathematics; Mathematics, general
ISSN
0003-889X
eISSN
1420-8938
DOI
10.1007/BF01199016
Publisher site
See Article on Publisher Site

Abstract

Arch. Math, Vol. 59, 65-79 (1992) 0003-889X/92/5901-0065 $ 4.50/0 9 1992 Birkh/iuser Verlag, Basel The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces By CHRISTIAN BAR O. Introduction. The spin structures of an oriented Riemannian homogeneous space M = G/H can be characterized by lifts of the isotropy representation. These basics are studied in the first section. In the second section we calculate the Dirac operator in algebraic terms, generalizing a well-known formula for symmetric spaces. Using Frobe- nius reciprocity, stated in Section 3, we calculate the Dirac spectrum of S 3 and quotients equipped with Berger metrics. Under the collapse onto S 2 some eigenvalues of S 3 con- verge to those of S 2 while the rest tends to + oe. The real projective space NP 3 equipped with the first spin structure shows the same behaviour whereas all eigenvalues of IIP 3 with the second spin structure tend to + oo. The Berger spheres also show that the dimension of the space of harmonic spinors highly depends on the Riemannian metric, an observation due to Hitchin, see [4]. This paper is part of the author's thesis, see [1]. 1. Spin structures on

Journal

Archiv der MathematikSpringer Journals

Published: Feb 4, 2005

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