Gaussian curvature - Knowlpedia

Let $(S,g)$ be a two-dimensional Riemannian manifold. Its Gaussian curvature is the function $K:S\to\mathbb R$ whose value at $p$ is the sectional curvature of the unique two-plane $T_pS$ :

K(p)=K_g(T_pS).

Curvature tensor formula

For an orthonormal basis $e_1,e_2$ of $T_pS$ , using the stated Riemann curvature tensor convention,

K(p)=g(R(e_1,e_2)e_2,e_1).

Intrinsic and extrinsic descriptions

The value $K(p)$ depends only on the metric near $p$ , not on an embedding of $S$ in Euclidean space. If $S$ is embedded as a regular surface in $\mathbb R^3$ , then $K$ is also the product of the two principal curvatures, with the usual induced metric.

Examples and consequences

The Euclidean plane has $K=0$ , the unit round sphere has $K=1$ , and the hyperbolic plane of curvature $-1$ has $K=-1$ . For a closed oriented Riemannian surface, the Chern–Gauss–Bonnet theorem gives

\frac{1}{2\pi}\int_S K\,\operatorname{vol}_g=\chi(S).

Sign convention

Changing the overall sign convention for the Riemann curvature tensor changes the sign of $K$ . The convention here is the one displayed in the Riemann curvature tensor and sectional curvature knowls.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. Publisher record. Relevant: Chapter 7.
Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976. Relevant: Chapters 4–5, Gaussian curvature and the Theorema Egregium.