Affine analogues of the Sasaki-Shchepetilov connection

Maria Robaszewska

Abstract


For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM⊕E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM⊕(R⨯M) and two on TM⊕(R2⨯M) are constructed. It is shown that two of those connections - one from each pair - may be identified with the standard flat connection in RN, after suitable local affine embedding of (M,∇) into RN.

Keywords


connection on a vector bundle; associated vector bundle; connection form; locally symmetric connection

Mathematics Subject Classification (2010)


53C07; 53B05; 53B15

References


Crittenden, Richard J. "Covariant differentiation." Quart. J. Math. Oxford Ser. (2) 13 (1962): 285-298.

Gancarzewicz, Jacek. Zarys współczesnej geometrii różniczkowej. Warszawa: Script, 2010.

Nomizu, Katsumi and Takeshi Sasaki. "Affine differential geometry. Geometry of affine immersions." Vol. 111 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 1994.

Nomizu, Katsumi and Barbara Opozda. "Locally symmetric connections on possibly degenerate affine hypersurfaces." Bull. Polish Acad. Sci. Math. 40, no. 2 (1992): 143-150.

Opozda, Barbara. "Locally symmetric connections on surfaces." Results Math. 20, no. 3-4 (1991): 725-743.

Opozda, Barbara. "Some relations between Riemannian and affine geometry." Geom. Dedicata 47, no. 2, (1993): 225-236.

Opozda, Barbara. "A characterization of affine cylinders." Monatsh. Math. 121, no. 1-2 (1996): 113-124.

Robaszewska, Maria. "On some flat connection associated with locally symmetric surface." Ann. Univ. Paedagog. Crac. Stud. Math. 13 (2014): 19-44.

Sasaki, Ryu. "Soliton equations and pseudospherical surfaces." Nuclear Phys. B 154, no. 2 (1979): 343-357.

Shchepetilov, Alexey V. "The geometric sense of the Sasaki connection." J. Phys. A 36, no. 13 (2003): 3893-3898.


Full Text: PDF

Download statistics: 1658

Licencja Creative Commons
This article by Maria Robaszewska is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej