Gauss Formula

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.

Content

Wording

Let be $S$ ${\ displaystyle S}$ - two-dimensional surface in three-dimensional Riemannian space $M$ ${\ displaystyle M}$ . Then

K_{S}(x)=K_{M}(\sigma _{S}(x))+\kappa _{1}(x)\kappa _{2}(x),

{\ displaystyle K_ {S} (x) = K_ {M} (\ sigma _ {S} (x)) + \ kappa _ {1} (x) \ kappa _ {2} (x),}

Where

$K_{S}$ ${\ displaystyle K_ {S}}$ - Gaussian surface curvature $S$ ${\ displaystyle S}$ at the point $x\in S$ ${\ displaystyle x \ in S}$ ,
$K_{M}(\sigma _{S}(x))$ ${\ displaystyle K_ {M} (\ sigma _ {S} (x))}$ - sectional curvature of space $M$ ${\ displaystyle M}$ in the direction $\sigma _{S}(x)$ ${\ displaystyle \ sigma _ {S} (x)}$ tangent to the surface $S$ ${\ displaystyle S}$ at the point $x$ ${\ displaystyle x}$ ,
$\kappa _{1}(x)$ ${\ displaystyle \ kappa _ {1} (x)}$ , $\kappa _{2}(x)$ ${\ displaystyle \ kappa _ {2} (x)}$ - main surface curvatures $S$ ${\ displaystyle S}$ at the point $x .$ ${\ displaystyle x.}$

Generalization to large dimensions

The formula can be generalized to an arbitrary dimension and codimension of an embedded submanifold $S\subset M$ ${\ displaystyle S \ subset M}$ . In this case, the curvature tensor $R_{S}$ ${\ displaystyle R_ {S}}$ submanifolds $S$ ${\ displaystyle S}$ expressed through the contraction of the curvature tensor $R_{M}$ ${\ displaystyle R_ {M}}$ of space $M$ ${\ displaystyle M}$ subspace tangent to $S$ ${\ displaystyle S}$ and the second quadratic form $q_{S}$ ${\ displaystyle q_ {S}}$ submanifolds $S$ ${\ displaystyle S}$ on tangent space $T S$ ${\ displaystyle TS}$ with values in normal space to $S$ ${\ displaystyle S}$ :

\langle R_{S}(X,Y)Z,W\rangle =\langle R_{M}(X,Y)Z,W\rangle +\langle q_{S}(Y,W),q_{S}(X,Z)\rangle -\langle q_{S}(X,W),q_{S}(Y,Z)\rangle .

{\ displaystyle \ langle R_ {S} (X, Y) Z, W \ rangle = \ langle R_ {M} (X, Y) Z, W \ rangle + \ langle q_ {S} (Y, W), q_ {S} (X, Z) \ rangle - \ langle q_ {S} (X, W), q_ {S} (Y, Z) \ rangle.}

^[one]

It should be borne in mind that different authors determine the curvature tensor with a different sign and order of arguments.

Notes

↑ Postnikov M.M. Rimanova, Geometry M .: Factorial, 1998, p. 337.

Literature

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.

[1] Postnikov M.M. Rimanova, Geometry M .: Factorial, 1998, p. 337.