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Conformal and Grassmann structures. (English) Zbl 0924.53025
In this survey, the main results of the real theory of (pseudo-)conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures and their differential prolongations are found. A complete system of geometric objects of the almost Grassmann structure totally defining its geometric structure is determined. The vanishing of these objects determines a locally Grassmann manifold. It is proved that the integrable almost Grassmann structures are locally Grassmann. The criteria of semiintegrability of almost Grassmann structures are proved in invariant form.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53A30 Conformal differential geometry (MSC2010)
Full Text: DOI
References:
[1] Akivis, M.A., Three-webs of multidimensional surfaces, (), 7-31, (in Russian). · Zbl 0178.24601
[2] Akivis, M.A.; Akivis, M.A., Webs and almost-Grassmann structures, Sibirsk. mat. zh., Siberian math. J., 23, 6, 763-770, (1982), English transl. · Zbl 0516.53013
[3] Akivis, M.A., On the differential geometry of a Grassmann manifold, Tensor (N.S.), 38, 273-282, (1982), (in Russian). · Zbl 0504.53010
[4] M.A. Akivis, Completely isotropic submanifolds of a four-dimensional pseudoconformal structure, Izv. Vyssh. Uchebn. Zaved. Mat.{\bf1983} (1) (248) 3-11 (in Russian). · Zbl 0526.53054
[5] Akivis, M.A., On the theory of conformal structures, (), 44-52, (in Russian). · Zbl 0844.53015
[6] Akivis, M.A.; Goldberg, V.V., Conformal differential geometry and its generalizations, (1996), Wiley-Interscience Publication New York · Zbl 0863.53002
[7] Akivis, M.A.; Konnov, V.V.; Akivis, M.A.; Konnov, V.V., Local aspects in conformal structure theory, Uspekhi mat. nauk, Russian math. surveys, 48, 1, 1-35, (1993), English transl. · Zbl 0804.53022
[8] Akivis, M.A.; Shelekhov, A.M., Geometry and algebra of multidimensional three-webs, (1992), Kluwer Dordrecht · Zbl 0792.53009
[9] Atiyah, M.F.; Hitchin, N.L.; Singer, I., Self-duality in four-dimensional Riemannian geometry, (), 425-461 · Zbl 0389.53011
[10] Bailey, T.N.; Eastwood, M.G., Complex paraconformal manifolds: their differential geometry and twistor theory, (), 61-103, (1) · Zbl 0728.53005
[11] Baston, R.J., Almost Hermitian symmetric manifolds. I. local twistor theory, Duke math. J., 63, 1, 81-112, (1991) · Zbl 0724.53019
[12] Cartan, É.; Cartan, É., LES sous-groupes des groupes continus de transformations, (), 25, 3, 719-856, (1908) · JFM 39.0206.04
[13] Cartan, É.; Cartan, É., Sur LES équations de la gravitation d’Einstein, (), 1, 549-611, (1922) · JFM 48.0993.02
[14] Cartan, É.; Cartan, É., Sur LES espaces conformes généralisés et l’univers optique, (), 174, 622-624, (1922) · JFM 48.0854.04
[15] Cartan, É.; Cartan, É., LES espaces à connexion conforme, (), 2, 747-797, (1923) · JFM 50.0493.01
[16] Dhooghe, P.F., Grassmannian structures on manifolds, Bull. belg. math. soc. Simon stevin, 1, 1, 597-622, (1994) · Zbl 0934.53025
[17] Eisenhart, L.P., Riemannian geometry, (1926), Princeton Univ. Press Princeton, N.J · Zbl 0041.29403
[18] 6th printing, 1966.
[19] Gardner, R., The method of equivalence and its applications, (1989), SIAM Philadelphia, PA
[20] Goldberg, V.V.; Goldberg, V.V., (n + 1)-webs of multidimensional surfaces, Dokl. akad. nauk SSSR, Soviet math. dokl., 14, 3, 795-799, (1973), English transl. · Zbl 0304.53017
[21] V.V. Goldberg, The almost Grassmann manifold that is connected with an (n + 1)-web of multidimensional surfaces, Izv. Vyssh. Uchebn. Zaved. Mat.{\bf1975} (8) (159) 29-38 (in Russian).
[22] Goldberg, V.V., Theory of multidimensional (n + 1)-webs, (1988), Kluwer Dordrecht · Zbl 0668.53001
[23] Goncharov, A.B., Generalized conformal structures on manifolds, Selecta math. soviet., 6, 306-340, (1987) · Zbl 0632.53038
[24] Hangan, Th., Géométrie différentielle grassmannienne, Rev. roumaine math. pures appl., 11, 5, 519-531, (1966) · Zbl 0163.43402
[25] Hangan, Th., Tensor-product tangent bundles, arch. math. (basel), 19, 4, 436-440, (1968) · Zbl 0172.47001
[26] Hangan, Th., Sur l’intégrabilité des structures tangentes produits tensoriels réels, Ann. mat. pura appl., 126, 4, 149-185, (1980) · Zbl 0457.53016
[27] T. Ishihara, On tensorproduct structures and Grassmannian structures, J. Math. Tokushima Univ.{\bf1972} (4) 1-17. · Zbl 0218.53049
[28] Yu.I. Mikhailov, On the structure of almost Grassmannian manifolds, Izv. Vyssh. Uchebn. Zaved. Mat.{\bf1978} (2) 62-72 (in Russian).
[29] Rosenfeld, B.A., Quasielliptic spaces, (), 49-70, (in Russian).
[30] Sternberg, S.; Sternberg, S., Lectures on differential geometry, (1983), Wiley-Interscience Publication New York, N.Y · Zbl 0521.58033
[31] Weyl, H., Reine infinitesimalgeometrie, Math. Z., 2, 384-411, (1918) · JFM 46.1301.01
[32] Yano, K., Sur LES equations de codazzi dans la géométrie conforme des espaces de Riemann, (), 340-344 · JFM 65.0798.01
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