On the spectrum and the geometry of spherical space forms. I
In this paper connected compact manifolds M of constant sectional curvature one are studied. Using the method of the Euler-Poisson-Darboux equation an explicit Poisson formula for the distribution \(\zeta_ n:=\sum\cos \sqrt{\mu +(n-1)^ 2/4}\cdot, n:=\dim M\) was derived. Here the sum runs through all \(\mu \in spec(M)\). This formula describes the relation between \(\zeta_ n\) and geometrical data of M. Furthermore the lengths of closed geodesics belonging to a free homotopy class are calculated. At the end of the paper a short proof of the statement \(\sin g \sup p \zeta_ n=\{0\}\cup L(M)\) is given, where L(M) is the set of all periods of the periodic geodesics c of M normed by \(\|\dot c\| =1\).
scientific article; zbMATH DE number 3872349