Mal'cev's criterion

Short description: Theorem for nilmanifolds

In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational.

The relevance of Mal'cev's criterion comes from the fact that it gives us a one-to-one correspondence between isomorphism classes of Lie algebras with rational structure constants and compact nilmanifolds. Indeed, Mal'cev showed that compact nilmanifolds are precisely quotients of simply connected nilpotent Lie groups by a lattice.

Relevance in Kähler geometry

Mal'cev's criterion is relevant in Kähler geometry because compact nilmanifolds with a Kähler structure must be diffeomorphic to a torus. Therefore, when looking for manifolds that do not admit a Kähler structure, one may use Mal'cev's criterion to generate a compact nilmanifold from any rational Lie algebra. By constructing other structures on the manifold, such as a complex or symplectic structure, one finds a non-Kähler manifold of said structure. It therefore helps finding solutions to the Thurston–Weinstein problem, which concerns itself with the existence of non-Kähler symplectic manifolds.

References

• Benson, Chal; Gordon, Carolyn S. (1988). "Kähler and symplectic structures on nilmanifolds". Topology 27 (4): 513–518. doi:10.1016/0040-9383(88)90029-8. ISSN 0040-9383. http://dx.doi.org/10.1016/0040-9383(88)90029-8. 
• Bland, John; Kim, Kang-Tae; Krantz, Steven George, eds (2003). "The moduli space of 2-step nilpotent Lie algebras of type (p,q)". Explorations in Complex and Riemannian Geometry: A Volume Dedicated to Robert E. Greene. Contemporary. 332. American Mathematical Society. pp. 37–72. ISBN 9780821832738. 
• Brin, Michael; Hasselblatt, Boris; Pesin, Ya. B., eds (2004). "Geometry of 2-step nilpotent Lie groups with a left invariant metric". Modern Dynamical Systems and Applications. Cambridge University Press. pp. 67–102. ISBN 9780521840736. 
• Hasegawa, Keizo (1989). "Minimal models of nilmanifolds". Proceedings of the American Mathematical Society 106 (1): 65–71. doi:10.1090/s0002-9939-1989-0946638-x. ISSN 0002-9939. http://dx.doi.org/10.1090/s0002-9939-1989-0946638-x. 

Further reading

• Mal'cev, Anatoly Ivanovich (1962). "On a class of homogeneous spaces.". American Mathematical Society Translations Series One 9: 276–307.  also —
  • Mal'cev, Anatoly Ivanovich (1949). "On a class of homogeneous spaces.". Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya (13): 9–32.