A bumpy metric theorem and the Poisson relation for generic strictly convex domains
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 \(\Omega\)).
scientific article; zbMATH DE number 4173937
scientific article