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An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

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Abstract

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.

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REFERENCES

  1. Yu. A. Aminov, “An expression for the Riemann tensor of a submanifold given by a system of equations in the Euclidean space,” Mat. Zametki [Math. Notes] (to appear).

  2. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N.J., 1997.

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  3. Z. P. Dombrowski, “Krümmungsgrößen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten,” Math. Nachr., 38 (1968), no. 3/4, 133-180.

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Authors and Affiliations

  1. Kharkov State University, Ukraine

    Yu. A. Aminov

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  1. Yu. A. Aminov
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Aminov, Y.A. An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space. Mathematical Notes 72, 595–599 (2002). https://doi.org/10.1023/A:1021476702275

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  • DOI: https://doi.org/10.1023/A:1021476702275

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