Search
Submit an article RU
Differential Geometry of Manifolds

Current issue

Back to the list Download the article
Kuleshov A.V.

On the structure forms of a projective structure

DOI
10.5922/0321-4796-2022-53-7
Pages
68-83
smooth manifold quotient frame frame bundle projective structure differential geometric structure structure forms structure equa­tions

Abstract

A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener­ates the sequence of differential geometric structures. The construction is following:

Step 1. For a smooth manifold the so-called quotient frame bundle as­sociated to the 2nd order frame bundle on the manifold is constructed.

Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are con­structed. These mappings are the differential geometric structures.

Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the pro­jective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.

Reference

1.  Evtushik, L. E., Lumiste, Yu. G., Ostianu, N. M., Shirokov, A. P.: Dif­fe­rential-geometric structures on manifolds. J. Soviet Math., 14:6, 1573—1719 (1980).

2.  Kuleshov, A. V.: Center-projective frames as equivalence classes of the second order frames. DGMF. Kaliningrad. 46, 94—107 (2015).

3.  Ovsienko, V., Tabachnikov S.: Projective differential geometry. Old and new. Cambridge (2005).

4.  Eisenhart, L. P.: Non-Riemannian Geometry. New York (1927).

5.  Whitehead, J. M. C.: Locally homogeneous spaces in differential geometry. Ann. Math. 33:4, 681—687 (1932).

6.  Ehresmann, C.: Sur les espaces localement homogènes. L’Enseign. Math. 35, 317—333 (1936).

7.  Choi, C., Goldman, W.: The classification of real projective struc­tures on compact surfaces. Bull. Am. Math. Soc. 34, 161—171 (1997).

8.  Cooper, D., Goldman, W.: A 3-Manifold with no Real projective structure. http://arxiv.org/abs/1207.2007.

9.  Ovsienko, V. Yu.: Projective structure and contact forms. Funct. anal. and its app., 28:3, 187—197 (1994).

10.  Kobayashi, S.: Transformation groups in differential geometry. Mos­cow (1986).

11.  Kuleshov, A. V.: On the equivalence of two viewpoints on the cen­ter-projective frames. DGMF. Kaliningrad. 48, 60—65 (2017).