Mathematical Physics Seminar

    Thursday 05.XI.2009

    Time: 15:15-16:15

    Room: 431a
    Instytut Fizyki, ul. Reymonta 4, 30-059 Kraków

    Dmitri Alekseevsky (University of Edinburgh, UK)

    Para-Kähler-Einstein homogeneous manifolds of a semisimple Lie group

    Abstract:
    A 2n-dimensional pseudo-Riemannian manifold (M, g) is called para-Kähler if it admits a
    parallel para-complex structure K that is an involutive field of endomorphisms or, equivalently,
    two complementary n-dimensional isotropic parallel distributions L±. A para-Kähler manifold
    can also be described as a symplectic manifold with symplectic form ω = g ° K and two
    complementary integrable Lagrangian distributions L±. We give a description of homogeneous
    para-Kähler manifolds of a real semisimple Lie group G in terms of its crossed Satake diagrams
    and invariant symplectic structures. The main result is a classification of invariant para-Kähler-
    -Einstein metrics on homogeneous manifolds M = G/H of a semisimple Lie group G in terms of
    Koszul forms. This is a para-complex analogue of a classification of invariant Kähler-Einstein
    metrics on compact homogeneous manifolds, given in a joint work with A.M. Perelomov. The
    talk is based on joint works with C. Medori and A. Tomassini.

    References:
    [1] D.V. Alekseevsky, C. Medori, A. Tomassini: "Homogeneous para-Kähler Einstein manifolds",
    arXiv:math.DG/0806.2272
    [2] D.V. Alekseevsky, C. Medori, A. Tomassini: "Maximally homogeneous para-CR manifolds
    of semisimple type", arXiv:math.DG/0808.0431


    (Please notice the exceptional time and venue this week.)


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