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Riemannian nilmanifolds and the trace formula


Author: Ruth Gornet
Journal: Trans. Amer. Math. Soc. 357 (2005), 4445-4479
MSC (2000): Primary 35P20, 58J53, 53C22; Secondary 58J40, 22E25
DOI: https://doi.org/10.1090/S0002-9947-05-03965-6
Published electronically: June 10, 2005
MathSciNet review: 2156717
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Abstract: This paper examines the clean intersection hypothesis required for the expression of the wave invariants, computed from the asymptotic expansion of the classical wave trace by Duistermaat and Guillemin. The main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary Riemannian two-step nilmanifold to satisfy the clean intersection hypothesis. This condition is stated in terms of metric Lie algebra data. We use the calculation to show that generic two-step nilmanifolds satisfy the clean intersection hypothesis. In contrast, we also show that the family of two-step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two-step nilmanifolds. Finally, we give examples of nilmanifolds that fail the clean intersection hypothesis.


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Additional Information

Ruth Gornet
Affiliation: Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019-0408
Email: rgornet@uta.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03965-6
Keywords: Isospectral manifolds, nilmanifolds, length spectrum, trace formula
Received by editor(s): August 28, 2003
Published electronically: June 10, 2005
Additional Notes: This material is based in part on work supported by the Texas Advanced Research Program under Grant No. 003644-002 and by NSF grants DMS-9753220 (preliminary) and DMS-0204648
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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