Quasihomogeneous manifolds and generalized almost-Hermitian structures

V. F. Kirichenko

Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 473-486.

Résumé

The author considers new classes of differential-geometric structures on smooth manifolds that generalize almost-Hermitian and almost-contact $f$-structures and others, and establishes the local structure of generalized homogeneous manifolds that carry such a structure. Bibliography: 24 titles.

Bibliographie

[1] Ambrose W., Singer J. M., “On homogeneous Riemannian manifolds”, Duke Math. J., 25 (1958), 647–669 | DOI | MR | Zbl

[2] Kirichenko V. F., “Ob odnorodnykh rimanovykh prostranstvakh s invariantnoi tenzornoi strukturoi”, Dokl. AN SSSR, 252:2 (1980), 291–293 | MR | Zbl

[3] Likhnerovich A., Teoriya svyaznostei v tselom i gruppy golonomii, IL, M., 1960

[4] Gray A., “Riemannian manifolds with geodesic symmetries of order 3–7”, J. Different. Geom., 7:3 (1972), 343–369 | MR | Zbl

[5] Gray A., “Nearly Kähler manifolds”, J. Different. Geom., 4:3 (1970), 283–309 | MR | Zbl

[6] Gray A., “The structure of nearly Kahler manifolds”, Math. Ann., 223:3 (1976), 233–248 | DOI | MR | Zbl

[7] Barros M., Naveira A., “Decomposition des variétés presque kählériennes verifiant la deuxième condition de courbure”, C.R. Acad. Sci., 284:22 (1977), A1461–A1463 | MR

[8] Barros M., Ramirez A., “Decomposition of quasi Kähler manifolds which satisfy the first curvature condition”, Demonstr. Math., 11:3 (1978), 685–694 | MR | Zbl

[9] Kirichenko V. F., “$K$-prostranstva maksimalnogo ranga”, Matem. zametki, 22:4 (1977), 465–476 | MR | Zbl

[10] Yano K., “On a structure defined by a tensor field $f$ of type $(1,1)$ satisfying $f^3+f=0$”, Tensor, 14 (1963), 99–109 | MR | Zbl

[11] Singh K. D., Singh Rakeshwar, “Some $f(3,\varepsilon)$-structure manifolds”, Demonstr. Math., 10:3,4 (1977), 637–645 | MR | Zbl

[12] Hervella L., Vidal E., “Nouvelles géométries pseudo-kähleriennes. $G_1$ et $G_2$”, C.R. Acad. Sci., 283:3 (1976), A115–A118 | MR

[13] Gray A., Hervella L., “The sixteen classes of almost Hermitian manifolds and their linear invariants”, Ann. Mat. Pura et Appl., 123 (1980), 35–58 | DOI | MR | Zbl

[14] Sasaki S., “On differentiable manifolds with certain structures which are closely related to almost contact structure, I”, Tohoku Math. J., 12:3 (1960), 459–476 | DOI | MR | Zbl

[15] Komrakov B. P., “Odnorodnye prostranstva s invariantnymi pochti kontaktnymi strukturami”, Tr. seminara po vekt. i tenz. analizu, 18, 1978, 264–293 | MR

[16] Dube K. K., “On almost hyperbolic Hermitian manifolds”, Analele Univ. Timisoara Seria Stinite Math., 11, no. 1, 1973, 47–54 | MR | Zbl

[17] Rosca R., “Sous-variétés anti-invariantes d'une variété parakakählerienne structurée par une connexion géodésique”, C.R. Acad. Sci., AB287:7 (1979), A539–A541 | MR

[18] Upadhyay M. D., Dube K. K., “Almost hyperbolic contact $(f,g,\eta,\xi)$-structure”, Acta Math., Acad. Scient., Hung., XVII (1973), 13–15 | MR

[19] Adati T., Miyaszawa T., “On paracontact Riemannian manifolds”, TRU Math., 13:2 (1977), 27–39 | MR | Zbl

[20] Kim Jin Bai., “Notes on $f$-manifolds”, Tensor, 29:3 (1975), 299–302 | MR | Zbl

[21] Awasti C. P., Gupta V. C., “Integrability conditions of a $F(k,k-2)$-structure satisfying $F^k+F^{k+2}=0$”, Demonstr. Math., 10:3,4 (1977), 577–588 | MR

[22] Nakayama H., “On framed $f$-manifolds”, Kodai Math. Semin. Repts, 18:4 (1966), 293–306 | DOI | MR

[23] Kobayashi S., Nomizu K., Foundations of differential geometry, v. 1, 2, Interscience, New York, 1963, 1969 | MR | Zbl

[24] Takamatsu K., Watanabe Y., “Classification of conformally flat $k$-space”, Tohoku Math. J., 24:3 (1972), 435–440 | DOI | MR | Zbl