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 Izv. Vyssh. Uchebn. Zaved. Mat., 2010, Number 11, Pages 63–73 (Mi ivm7151)

Dual Riemannian spaces of constant curvature on a normalized hypersurface

A. V. Stolyarov

Chair of Geometry, Chuvash State Pedagogical University, Cheboksary, Russia

Abstract: In this paper we obtain the following results: 1) we prove that in a differential neighborhood of the fourth order a regular hypersurface $\mathrm V_{n-1}$ embedded in a projective-metric space $\mathrm K_n$, $n\geqslant3$, intrinsically induces the dual projective-metric space $\overline K_n$; 2) we obtain an invariant analytical condition under which the normalization of the hypersurface $\mathrm V_{n-1}\subset\mathrm K_n$ (the tangential hypersurface $\overline{\mathrm V}_{n-1}\subset\overline{\mathrm K}_n$) by fields of quasitensors $H^i_n$, $H_i$ ($\overline H^i_n$, $\overline H_i$) induces a Riemannian space of constant curvature. Note that when these two conditions are fulfilled simultaneously, spaces $R_{n-1}$ and $\overline R_{n-1}$ are dual with the same identical constant curvature $\mathrm K=-\frac1c$; 3) we give geometric descriptions of the obtained analytical conditions.

Keywords: projective-metric space, duality, normalization, Riemannian connection, Riemannian space of constant curvature.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:11, 56–65

Bibliographic databases:

UDC: 514.576

Citation: A. V. Stolyarov, “Dual Riemannian spaces of constant curvature on a normalized hypersurface”, Izv. Vyssh. Uchebn. Zaved. Mat., 2010, no. 11, 63–73; Russian Math. (Iz. VUZ), 54:11 (2010), 56–65

Citation in format AMSBIB
\Bibitem{Sto10} \by A.~V.~Stolyarov \paper Dual Riemannian spaces of constant curvature on a~normalized hypersurface \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2010 \issue 11 \pages 63--73 \mathnet{http://mi.mathnet.ru/ivm7151} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2814565} \transl \jour Russian Math. (Iz. VUZ) \yr 2010 \vol 54 \issue 11 \pages 56--65 \crossref{https://doi.org/10.3103/S1066369X1011006X} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78649529851}