The Lichnerowicz conjecture on harmonic manifolds
There are many equivalent definitions for harmonic spaces. For example, a Riemannian manifold is said to be locally harmonic if and only if for each point \(m\in M\) the volume density function \(\theta_ m\) of the exponential map \(\exp_ m\) centered at m is constant on sufficiently small geodesic spheres or equivalently, if and only if the mean curvature of all sufficiently small geodesic spheres \(G_ m(r)\), with center m and radius r, is constant on \(G_ m(r)\). There is a corresponding notion of a globally harmonic space. All manifolds which are locally isometric to a two-point homogeneous space are trivially harmonic and it is a longstanding conjecture of A. Lichnerowicz that these spaces are the only examples. A positive answer was known only for dimensions \(\leq 4.\) This important paper contains the most striking result in this domain: Any simply connected compact harmonic space is a two-point homogeneous space. At the same time the author gives a new proof of the fact that any two-point homogeneous space is symmetric. The author uses extensively the compactness condition and, unfortunately, the method cannot be used to give an answer in the general case, not even for complete spaces. So, further resarch is needed to obtain a complete answer for the conjecture of Lichnerowicz.
scientific article; zbMATH DE number 4124533