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Differential Geometry of Manifolds

2023 №54(2)

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Shevchenko Yu.I. Vyalova A.V.

Linear and projective connections over a smooth manifold

DOI
10.5922/0321-4796-2023-54-2-8
Pages
78-91
bundle of coframes linear connection torsion and curva­tu­re tensors weak and strong projective connection classic projective connection

Abstract

The principal bundles of the first order coframes and the second order coframes, as well as factor bundle of centroprojective (coaffine) coframes are considered. In the bundle of linear coframes a connection is given with the help of the field of connection object. The torsion and curvature tensors of this linear connection are determined. Special connections are singled out: torsion-free, curvature-free. The space of a linear connection devoid of torsion and curvature is an affine group, that served as the basis for classical name «affine connection».

Under the specializations of a manifold, strong and weak projectivity conditions was introduced, which make it possible to single out the cof­ra­me bundles. The connections in these principal bundles are called strong and weak projective connections.

In the case of symmetric linear connection, when the torsion is absent, the object of classic projective connection is considered. Connec­tion forms are introduced and their structure equations are found. Hence it follows that classic projective connection is neither fundamental-group nor linear differential-geometric. It is proved, that the curvature object of this connection forms a quasitensor together with the connection object only. It is shown, that classic projective connection degenerates into diffe­rent from the original linear connection on the image of a section of some homogeneous bundle.



Reference

1. Laptev, G. F.: Group-theoretic method in differential geometric investigation. Tr. 3rd All-Union Math. Congr., 3, 409—418 (1958).

2. Laptev, G. F.: Fundamental infinitesimal structures of higher orders on a smooth manifold. Tr. Geom. Sem., 1, 139—189 (1966).

3. Laptev, G. F.: Manifolds, imbedded in generalized spaces. Tr. 4th All-Union Math. Congr., 1961, 2. Leningrad, 226—233 (1964).

4. Evtushik, L. E., Lumiste, Yu. G., Ostianu, N. M., Shirokov, A. P.: Differential-geometric structures on manifolds. Itogi Nauki i Tekhn. Probl. Geom., 9 (1979).

5. Norden, A. P.: Spaces with an affine connection. Moscow (1976).

6. Veblen, O.: Generalized projective geometry. J. Lond. Math. Soc., 4, 140—160 (1929).

7. Gordeeva, I.: Six classes of non-symmetric metric connections. DGMF, 38, 33—38 (2007).

8. Shevchenko, Yu. I., Vyalova A. V.: Metrics of a space with linear con­nection, which isn’t semisymmetric. DGMF, 53, 148—160 (2022).

9. Vagner, V. V.: The theory of composite manifolds. Tr. Sem. Vect. and Tenz. Analysis, 8, 11—72 (1950).

10. Shevchenko, Yu. I.: Holonomic and semi-holonomic submanifolds of smooth manifolds. DGMF, 46, 168—177 (2015).