Extending an idea of A. Gray: homogeneous k-symmetric spaces and generalized Hermitian geometry
by
Vitaly V. Balashchenko
Belarusian State University, Minsk, Belarus

Oral Communication

1. Hermitian geometry and 3-symmetric spaces.

Homogeneous spaces of Lie groups are of great importance in differential geometry. In particular, homogeneous spaces defined by Lie group automorphisms F provided the remarkable examples for main classes in Hermitian geometry [WG]. The most important role belongs to Riemannian 3-symmetric spaces (i.e. F³=id) that in naturally reductive case are homogeneous nearly Kähler manifolds [Gr]. The key point here is the existence of the canonical almost complex structure defined by the automorphism F of order 3 (see [S], [WG]).
2. Generalized Hermitian geometry.

Intensive investigations of almost contact structures, f-structures by K.Yano (f³+f=0) and some similar structures naturally led to a general concept of generalized Hermitian geometry constructed in 80-th (see, for example, [Ki]). The basic notion in the subject is a generalized almost Hermitian structure (GAH-structure) of any finite rank r. As to a metric f-structure, it is the most important example of GAH-structure of rank 1 and a natural generalization of AH-structure J. Constructing some main classes of GAH-structures is based on the properties of the so-called adjoint natural Q-algebra. We will formulate all the definitions with respect to a metric f-structure. For example, if the adjoint Q-algebra is anticommutative or abelian the corresponding metric f-structure is called f-structure of class G₁ or Hermitian f-structure respectively. The defining property for a Kähler f-structure can be written in the form Ñf=0. It should be mentioned that in particular case f=J all the notions above coincide with the similar those of Hermitian geometry. At last, we recall two recent generalizations of nearly Kähler structures. Namely, a metric f-structure is called Killing if Ñ_X(f)X=0 [Gri], and f is a nearly Kähler f-structure if Ñ_fX(f)f X=0 [B1], [B2].
3. Canonical structures on k-symmetric spaces.

In the theory of homogeneous F-spaces G/H (generalized symmetric spaces [Ko]) the structures generated by the automorphism F in a certain canonical way are of special interest. However for a long period practically the only example of such a structure was well-known. This is the canonical structure J on 3-symmetric spaces [WG], [S], [Gr]. It turns out that any regular F-space G/H possesses a commutative algebra of invariant canonical structures [BS]. Note the algebra is trivial in symmetric case (F²=id). The most important canonical structures (almost product P, almost complex J, f-structures by K. Yano) are completely described. In particular, for k-symmetric spaces (F^k=id) the explicit formulae for all the structures mentioned were indicated (see [BS]). For example, there is a unique (up to sign) canonical f-structure on 4-symmetric spaces. As to 5-symmetric spaces, they generally possess two canonical structures J₁ and J₂ as well as two canonical f-structures f₁ and f₂.
4. Canonical f-structures in generalized Hermitian geometry.

In contrast to Hermitian geometry, invariant structures have not been observed in generalized Hermitian geometry. The main result is that canonical f-structures on k-symmetric spaces present large classes of homogeneous f-manifolds in generalized Hermitian geometry. In particular, the general method for constructing nearly Kähler f-structures on the basis of canonical ones is obtained. The set of homogeneous Hermitian f-manifolds was also indicated [B3]. It should be noted that k-symmetric spaces in the cases k=4, 5 are of especial interest. For example, the following result was proved:
Theorem. Let G/H be a naturally reductive 4- or 5-symmetric space, f any of canonical f-structures on G/H. Then f is both a nearly Kähler and Hermitian f-structure. Moreover, the following conditions are equivalent: 1) f is a Kähler f-structure; 2) f is a Killing f-structure; 3) f is a quasi-Kähler f-structure; 4) f is an integrable f-structure; 5) G/H is a locally symmetric space.

Note that some results mentioned are just the extending of the classical ones due to A.Gray, J.Wolf, V.F.Kirichenko and other authors.
5. Examples.

Many classes of examples are provided with the classification of Riemannian 4- and 5-symmetric spaces of classical compact Lie groups (see [J], [TX]). Further, the 6-dimensional generalized Heisenberg group (N, g) admits the structure of Riemannian 4-symmetric space that is not naturally reductive [TV]. The canonical f-structure on (N, g) is both nearly Kählerian and Hermitian, but it is not Killing, Kählerian, and integrable. Some other well-known examples due to O.Kowalski [Ko] were also considered in the sense.

References

[BS] Balashchenko, V.V., Stepanov, N.A., Canonical affinor structures of classical type on regular F-spaces // Sbornik: Mathematics, 186(1995), no.11, 1551-1580.

[B1] Balashchenko, V.V., Riemannian geometry of canonical structures on regular F-spaces // Preprint No.174/1994, Fakultät für Mathematik der Ruhr-Universität Bochum, 1994, 1-19.

[B2] Balashchenko, V.V., Naturally reductive Killing f-manifolds // Russian Math. Surveys. 54(1999), no.3.

[B3] Balashchenko, V.V., Homogeneous Hermitian f-manifolds // Russian Math. Surveys (to appear).

[Gr] Gray A. Riemannian manifolds with geodesic symmetries of order 3 // J. Diff. Geom. 7(1972), no. 3-4, 343-369.

[Gri] Gritsans, A.S., Geometry of Killing f-manifolds // Russian Math. Surveys. 45(1990), 168-169.

[J] Jimenez, J.A., Riemannian 4-symmetric spaces // Trans. Amer. Math. Soc., 306(1988), no.2, 715-734.

[Ki] Kirichenko, V.F., Methods of generalized Hermitian geometry in the theory of almost contact manifolds // Problems of Geometry. V.18. Itogi Nauki i Tekniki. VINITI. Moscow. 1986, 25-71 (in Russian).

[Ko] Kowalski, O., Generalized symmetric spaces, LN in Math., V.805, Berlin, Heidelberg, New York: Springer-Verlag, 1980.

[S] Stepanov, N.A., Homogeneous 3-cyclic spaces // Soviet Math. (Iz. VUZ) 11(1967), no. 12.

[TV] Tricerri, F., Vanhecke, L., Homogeneous structures on Riemannian manifolds, London Math. Soc., LN Ser., no.83, 1983.

[WG] Wolf, J., Gray, A., Homogeneous spaces defined by Lie group automorphisms // J. Diff. Geom. 2(1968), no. 1-2, 77-159.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-62.