The Poisson formula for Euclidean space groups and some of its applications. II: The Jacobi transformation for flat manifolds
The paper is a continuation of a previous result of the author [ibid. 1, No.1, 13-23 (1982; Zbl 0518.58010)]. Let (M,g) be a compact flat Riemannian manifold and let \(\omega\) be the set of non-trivial free homotopy classes of closed curves in (M,g). For any \(P\in M\) and any closed geodesic c through P we denote by \(\Pi_{P,c}\) the linear mapping of \(M_ P\) onto itself induced by the parallel displacement along c. If \(c\in \vartheta\), \(P\in f_{\vartheta}(M_{\vartheta})\) then \(\Pi_{P,c}\) splits into mappings \(\Pi_{P,c}^{\tan g}\) and \(\Pi^{\perp}_{P,c}\) which are tangential and orthogonal to \(f_{\vartheta}(M(\vartheta))\); where \(\vartheta\in \omega\). Theorem 1. Claim (a): To every \(\vartheta\in \omega\) there belongs a compact flat manifold M(\(\vartheta)\) of dimension n(\(\vartheta)\), \(1\leq n(\vartheta)\leq n\), and a non-trivial free homotopy class \(\vartheta_ 0\) on M(\(\vartheta)\). Through each point \(P\in M(\vartheta)\) go closed geodesics belonging to \(\vartheta_ 0\). All of them have equal lengths \(\ell (\vartheta)\); their tangential vectors form K(\(\vartheta)\) parallel fields on M(\(\vartheta)\). Claim (b): There is an isometric immersion \(f_{\vartheta}: M(\vartheta)\to M\), such that \(f_{\vartheta}(M(\vartheta))\) is totally geodesic in (M,g). The closed geodesics of (M,g) belonging to \(\vartheta\) are exactly the curves \(f_{\vartheta}(c')\), where c' is any closed geodesic of M(\(\vartheta)\) belonging to \(\vartheta_ 0\). Claim (c): \(\Pi_{P,c}^{\tan g}=Id\); further \(D(\vartheta):=| Det(\Pi^{\perp}_{P,c}-Id)|\) is different from zero and does not depend on the choice of \(P\in f_{\vartheta}(M(\vartheta))\) and \(c\in \vartheta\). In Theorem 2 is established a Jacobi transformation formula. Further on the author expresses the traces of the heat kernel and the wave equation kernel of a compact flat Riemannian manifold by data which are connected with the behaviour of its closed geodesics.
scientific article; zbMATH DE number 3936019