The equivalent notion for a pseudo-Riemannian manifold is: Sectional curvature for a pseudo-Riemannian manifold
A Riemannian manifold viz a differential manifold equipped with a Riemannian metric .
Let be a tangent plane to at a point . Then, the sectional curvature of at is defined as follows: take two linearly independent vectors and in , and calculate:
viz the inner product of and with respect to .
Divide this by the square of the area of the parallelogram formed by and . This ratio defines the sectional curvature of , denoted as .
Here is the more explicit formula:
Note that the denominator cannot vanish because and are independent vectors.
For a pseudo-Riemannian manifold
Further information: Sectional curvature for a pseudo-Riemannian manifold
We can also define the sectional curvature of a pseudo-Riemannian manifold. The same definition works.
Related notions of curvature
- Ricci curvature associates a real number to every tangent direction at every point
- Scalar curvature assocaites a real number to every point
Related metric properties
- Riemannian manifold with positively lower-bounded curvature
- Positively curved Riemannian manifold
- Quasi-positively curved Riemannian manifold
- Nonnegatively curved Riemannian manifold
- Negatively curved Riemannian manifold
- Constant-curvature metric
- Flat metric
Sectional curvature determines Riemann curvature tensor
The following facts are true about the Riemann curvature tensor:
- The Riemann curvature tensor can be viewed as a -tensor:
- The above map is antisymmetric in (For full proof, refer: Antisymmetry of Riemann curvature tensor and antisymmetric in and . Further it is symmetric in the pairs and . Hence, it is a symmetric bilinear form on .
Thus, given the sectional curvature for all tangent planes, we can back-calculate the Riemann curvature tensor from it by the usual polarization trick. The idea is to view the Riemann curvature tensor as a symmetric bilinear form on , and the sectional curvature as the values that this form takes on pairs of (special kinds of) unit vectors in .