Lichnerowicz conjecture

In mathematics, the Lichnerowicz conjecture is a generalization of a conjecture introduced by Lichnerowicz (1944). Lichnerowicz's original conjecture was that locally harmonic 4-manifolds are locally symmetric, and was proved by Walker (1949). The Lichnerowicz conjecture usually refers to the generalization that locally harmonic manifolds are flat or rank-1 locally symmetric. It has been proven true for compact manifolds with fundamental groups that are finite groups (Szabó 1990) but counterexamples exist in seven or more dimensions in the non-compact case (Damek & Ricci 1992)

References

• Damek, Ewa; Ricci, Fulvio (1992), "A class of nonsymmetric harmonic Riemannian spaces", Bulletin of the American Mathematical Society, New Series, 27 (1): 139–142, arXiv:math/9207213, doi:10.1090/S0273-0979-1992-00293-8, MR 1142682
• Lichnerowicz, André (1944), "Sur les espaces riemanniens complètement harmoniques", Bulletin de la Société Mathématique de France, 72: 146–168, ISSN 0037-9484, MR 0012886
• Szabó, Z. I. (1990), "The Lichnerowicz conjecture on harmonic manifolds", Journal of Differential Geometry, 31 (1): 1–28, ISSN 0022-040X, MR 1030663
• Walker, A. G. (1949), "On Lichnerowicz's conjecture for harmonic 4-spaces", Journal of the London Mathematical Society, Second Series, 24: 21–28, doi:10.1112/jlms/s1-24.1.21, ISSN 0024-6107, MR 0030280