Abstract
The Gauss-Codazzi-Ricci equations governing the local isometric embedding of Riemannian spacesV n ⊂vn (N=n + P, P > 0) are interrelated by the Bianchi identities inV n andV N. This leads to redundancies which permit great simplification in the embedding problem, i.e. allows a neglect of part of the equations. By transcription, to the case of semi-Riemannian spaces, of a result of R. Blum we obtain a number of theorems and corollaries expressing forV n ⊂ VN this interdependency of the Gauss-Codazzi-Ricci equations. They form a generalization of previous results and are felt to be useful for the study of the geometrical properties of space-time and its three-dimensional space sections.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gupta, Y. K., and Goel, P. (1975).Gen. Rel. Grav.,5, 499.
Thomas, T. Y. (1936).Acta Math.,67, 169.
Blum, R. (1945/46).Bull. Math. Soc. Rumaine Sci.,47 (1–2), 144.
Blum, R. (1947).C. R. Acad. Sci. Paris,244, 708.
Blum, R. (1955).Can. J. Math.,7, 445.
Kobayashi, S., and K. Nomizu (1969).Foundations of Differential Geometry (Interscience Publ., John Wiley and Sons, New York), Vol. 2, Chap. VII, Sec. 6.
Eisenhart, L. P. (1964).Riemannian Geometry (Princeton University Press, Princeton, New Jersey), Chap. IV.
Vranceanu, Gh. (1961).Vorlesungen über Differentialgeometrie (Akademie Verlag, Berlin), Teil 1, Kap. V, §9.
Verbizkii, L. L. (1950).Trudy. Seminar Vektor, Tenzor. Anal.,8, 425.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goenner, H.F. On the interdependency of the Gauss-Codazzi-Ricci equations of local isometric embedding. Gen Relat Gravit 8, 139–145 (1977). https://doi.org/10.1007/BF00770733
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00770733