A bumpy metric theorem and the Poisson relation for generic strictly convex domains
type: Item
instance of: scholarly article
title: A bumpy metric theorem and the Poisson relation for generic strictly convex domains
zbMATH Open document ID: 0713.58005
DOI: 10.1007/BF01446922
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164714
review text: The paper contains two results. The first one generalizes the bumpy metric theorem of Abraham (which asserts that for every \(C^{\infty}\)- manifold M there is a residual set of smooth metrics which are bumpy in the sense that every closed geodesic is nondegenerate): for generic submanifolds of \({\mathbb{R}}^ n\) without boundary the induced standard metric is a bumpy metric. The second result is a consequence of the first one: for generic bounded strictly convex domains \(\Omega \subset {\mathbb{R}}^ m\), \(m\geq 2\), the singular support of the tempered distribution \(\sigma (t)=\sum^{\infty}_{j=1}\cos \lambda_ jt\) (where \(\{\lambda^ 2_ j\}\) denotes the spectrum of the Dirichlet problem on \(\Omega\}\) coincides with the ``length spectrum'' (corresponding to all generalized periodic geodesics in \(\Omega\)).
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4173937
zbMATH Keywords: bumpy metric
zbMATH Keywords: generic bounded strictly convex domains
zbMATH Keywords: geodesics
author: Luchezar Stoyanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4173937