The Gauss formula (Gaussian relation, Gaussian equation) is an expression for the Gaussian curvature of a surface in three-dimensional Riemannian space in terms of the principal curvatures and the sectional curvature of the ambient space. In particular, if the ambient space is Euclidean, then the Gaussian curvature of the surface is equal to the product of the principal curvatures at this point.
Let be - two-dimensional surface in three-dimensional Riemannian space . Then
- - Gaussian surface curvature at the point ,
- - sectional curvature of space in the direction tangent to the surface at the point ,
- , - main surface curvatures at the point
Generalization to large dimensions
The formula can be generalized to an arbitrary dimension and codimension of an embedded submanifold . In this case, the curvature tensor submanifolds expressed through the contraction of the curvature tensor of space subspace tangent to and the second quadratic form submanifolds on tangent space with values in normal space to :
It should be borne in mind that different authors determine the curvature tensor with a different sign and order of arguments.
- Gauss Formula - Bonn
- Postnikov M.M. Rimanova, Geometry M .: Factorial, 1998, p. 337.
- 1. Postnikov M. M. Rimanova Geometry M .: Factorial, 1998, p. 337.
- 2. Kobayashi Sh. , Nomizu K. Fundamentals of differential geometry M .: Nauka, 1981, T. 2, p. 30.