Abstract
A projective space in which a linear group acts ineffectively allows one to construct the corresponding space with a Cartan projective connection. We show that the structural equations of the Cartan space allow one to obtain differential equations for the components of the projective curvature-torsion tensor. This tensor contains the torsion tensor, the extended torsion tensor, and the affine curvature-torsion tensor. An analog of the Bianchi identities is found. An algorithm for constructing structural equations of the space with a Cartan projective connection is formulated. Using a generalized algorithm, we construct the structural equations of the planar space with a projective connection, whose special cases are the ruled space with an Akivis projective connection, the point space with a Cartan projective connection, and its dual hyperplanar space with a projective connection. We also prove that the curvature-torsion tensor of the plane space with projective connection has three subtensors, one of which is an analog of the torsion tensor of the Cartan space.
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References
M. A. Akivis, “On isoclinic three-webs and their interpretation in a ruled projective connected space,” Sib. Mat. Zh., 15, No. 1, 3–15 (1974).
M. A. Akivis, Multidimensional Differential Geometry [in Russian], Kalinin (1977).
E. Cartan, Spaces of Affine, Projective, and Conformal connections [in Russian], Izd-vo Kazan. Univ., Kazan (1962).
L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, A. P. Shirokov, “Differential-geometric structures on manifolds,” Probl. Geom., 9, 3–248 (1979).
Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin–Heidelberg–New York (1972).
G. F. Laptev, “Differential geometry of embedded manifolds. A group-theoretic method of differential geometric investigations,” Tr. Mosk. Mat. Obshch., 2, 275–382 (1953).
G. F. Laptev, “Fundamental higher-order infinitesimal structures on smooth manifolds,” Tr. Geom. Semin. VINITI, 1, 139–189 (1966).
Yu. I. Shevchenko, “Structure of the equipment of a manifold of linear figures,” in: Proc. VI Baltic Geometric Conference, Tallinn (1984), pp. 136–137.
Yu. I. Shevchenko, “Centroprojective connection in the space with a Cartan projective connection,” Differ. Geom. Mnogoobr. Figur, 36 (2005), pp. 154–160.
Yu. I. Shevchenko, “Holonomic and semi-holonomic submanifolds of smooth manifolds,” Differ. Geom. Mnogoobr. Figur, 46 (2015), pp. 168–177.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 180, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor V. T. Bazylev. Moscow, April 22-25, 2019. Part 2, 2020.
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Shevchenko, Y.I., Skrydlova, E.V. Planar Spaces with Projective Connections. J Math Sci 276, 580–586 (2023). https://doi.org/10.1007/s10958-023-06781-8
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DOI: https://doi.org/10.1007/s10958-023-06781-8
Keywords and phrases
- space with a Cartan projective connection
- curvature-torsion tensor
- analog of Bianchi identities
- ruled space with a projective connection
- planar space with a projective connection