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Differential Geometry of Manifolds

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Bashashina K. V.

Glued linear connection on surface of the projective space

DOI
10.5922/0321-4796-2020-51-3
Pages
22-28
projective space surface glued bundle linear connection glued linear connection Cartan projective connection curvature tensor

Abstract

We consider a surface as a variety of centered planes in a multidi­mensional projective space. A fiber bundle of the linear coframes appears over this manifold. It is important to emphasize the fiber bundle is not the principal bundle. We called it a glued bundle of the linear coframes. A connection is set by the Laptev — Lumiste method in the fiber bundle. The ifferential equations of the connection object components have been found. This leads to a space of the glued linear connection. The expres­sions for a curvature object of the given connection are found in the pa­per. The theorem is proved that the curvature object is a tensor. A condi­tion is found under which the space of the glued linear connection turns into the space of Cartan projective connection. The study uses the Cartan — Laptev method, which is based on cal­culating external differential forms. Moreover, all considerations in the article have a local manner.

Reference

1. Bazylev, V.T.: The geometry of differentiable manifolds. Moscow (1989).
2. Evtuchik, L. E., Lumiste, Yu. G., Ostianu, N.M., Širokov, A.P.: Differential-geometric structures on manifolds. J. Soviet Math., 14:6, 1573—1719 (1980).
3. Shevchenko, Yu. I.: Laptev and Lumiste methods connectivity in
the main bundle. DGMF. Kaliningrad. 37, 179—187 (2006).
4. Stolyarov A. V., Glukhova T.N.: Conformal differential geometry
of framed manifolds. Cheboksary (2007).
5. Shevchenko, Yu.I.: About Cartan equipment. DGMF. Kaliningrad.
14, 107—110 (1983).