The canonical geodesic involution and harmonic spaces
Let (M,g) be an analytic Riemannian manifold. If one calculates the power-series expansion for the mean curvature of a geodesic sphere (with respect to the radius), it can be shown that the coefficients are not independent functions on M. In fact, each odd coefficient can be calculated from the previous coefficients and their derivatives. It is the aim of this paper to give an explicit recursion formula. One gets hence an alternative proof of the author's previous result stating that the ''odd harmonicity conditions'' are consequences of the ''even harmonicity conditions'' [see the book by \textit{H. S. Ruse}, \textit{A. G. Walker} and \textit{T. J. Willmore}, Harmonic spaces (1961; Zbl 0134.392)]. As a by-product, the author shows that the characteristic polynomial of the metric tensor calculated in normal coordinates is invariant with respect to the canonical involution. So far, this was known only for the last coefficient of this polynomial, i.e., for the volume density function [see Chapter VI in: \textit{A. Besse,} Manifolds all of whose geodesics are closed (1978; Zbl 0387.53010)].
scientific article; zbMATH DE number 3844544
scientific article