5 problems

Lichnerowicz's conjecture. There are no other simply connected harmonic manifolds.

Hotelling's characterization conjecture. Hotelling's theorem holds in $M$ if and only if $M$ is harmonic.

Finiteness conjecture for harmonic manifolds. There are no non-trivial deformations of harmonic manifolds; hence there should be only finitely many classes of harmonic manifolds.

Besse's conjecture. In the Riemannian context, being infinitesimal harmonic at every point should imply being infinitesimal harmonic.

A harmonic space is a Riemannian manifold whose volume density in geodesic polar coordinates depends only on the radius. A Riemannian manifold is locally flat if each point has a f…