Gauss, Peterson-Codazzi, and Ricci equations in nonholonomic frames

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2017.2.6508 Gauss, Peterson-Codazzi, and Ricci equations in nonholonomic frames Shapovalova L. N. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 2. Abstract: The isometric immersion of the \(n\)-dimensional pseudo-Riemannian manifold to an \(m\)-dimensional pseudo-Riemannian space of the constant curvature is under consideration. The manifold is assumed to be Hausdorff and orientable. Using the non-holonomic frames the author derived Gauss, Peterson-Codazzi, Ricci equations for \(C^2\) immersion of this manifold into \(m\)-dimensional pseudo-Riemannian space of constant curvature. The main result is obtained with the use of generalized external de Rham derivation. It is found that in this context the forms of connectivity, immersion and torsion have continuous generalized exterior derivations. Keywords: submanifold, immersion, nonholonomic frame, Gauss equation, Peterson-Codazzi equation, Ricci equation. Language: Russian Download the full text For citation: Shapovalova L. N. Gauss, Peterson-Codazzi, and Ricci equations in nonholonomic frames. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 49-57. DOI 10.23671/VNC.2017.2.6508 + References 1. Eisenhart L. P. Riemannian Geometry. Moscow: Izdatinlit, 1948. 316 p. (in Russian). 2. Klimentov S. B. Global formulation of the main theorem of the surface theory. Ukrainski geometricheski sbornik [Ukrainian Geometry Collection]. 1979, no. 22, pp. 64-81 (in Russian). 3. Borovski Yu. E. Pfaff systems with - coefficients and geometry applications. Siberian Math. J., 1988, vol. 24, no. 2, pp. 10-16 (in Russian). 4. Markov P. E. General analytical and infinitesimal deformations of the immersions. Izvestiya VUZov. Mathematics. 1997, no. 9(424), pp. 21-34 (in Russian). 5. Zulanke R. and Wintgen P. Differential Geometry and Bundles. Moscow: Mir. 1975. 352 p. (in Russian). 6. Postnikov M. M. Lie Groups and Algebras. Moscow: Nauka, 1982. 480 p. (in Russian). 7. Wolf J. A. Spaces of Constant Curvature. Moscow: Nauka, 1982. 480 p. (in Russian). 8. Finikov S. P. Kartan's External Forms Method in Differential Geometry. Moscow-Leningrad, 1948. 432 p. (in Russian). 9. De Rham. Differential Manifolds. Moscow: Izdatinlit, 1956. 250 p. (in Russian). 10. Borovski Yu. E. Completely integrable Pfaff's systems. Izvestiya VUZov, Mathematics. 1959, no. 3, pp. 35-38 (in Russian). 11. Markov P. E. About imbeddability of the metrics which are close to imbedded. Ukrainski geometricheski sbornik [Ukrainian Geometry Collection], 1992, no. 35, pp. 49-67 (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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