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

• Besse, Arthur L. (1978). Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 93. Appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. Berlin–New York: Springer-Verlag. doi:10.1007/978-3-642-61876-5. ISBN 3-540-08158-5. MR 0496885. Zbl 0387.53010.
• Chen, Bang-Yen (2000). "Riemannian submanifolds". Handbook of Differential Geometry, Volume I. Handbook of Differential Geometry. Vol. 1. Amsterdam: North-Holland. pp. 187–418. arXiv:1307.1875. doi:10.1016/S1874-5741(00)80006-0. ISBN 978-0-444-82240-6. MR 1736854. Zbl 0968.53002.
• 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, doi:10.24033/bsmf.1359, ISSN 0037-9484, MR 0012886
• Szabó, Z. I. (1990), "The Lichnerowicz conjecture on harmonic manifolds", Journal of Differential Geometry, 31 (1): 1–28, doi:10.4310/jdg/1214444087, 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
• Yau, Shing Tung (1982). "Problem section". In Yau, Shing-Tung (ed.). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ: Princeton University Press. pp. 669–706. doi:10.1515/9781400881918-035. ISBN 978-1-4008-8191-8. MR 0645762. Zbl 0479.53001.