Sectional curvature
Introduction
In the field of differential geometry, the term "sectional curvature" is a fundamental concept. It is a measure of the curvature of a Riemannian manifold, which is a type of smooth manifold equipped with a Riemannian metric. The sectional curvature provides a way to quantify the curvature of two-dimensional planes embedded within the manifold.
Definition
Given a Riemannian manifold (M, g), the sectional curvature K is a function that assigns to each two-dimensional plane in the tangent space at a point p in M a real number. More formally, if σ is a 2-plane in TpM, the tangent space at p, then the sectional curvature K(σ) is defined by the following equation:
K(σ) = R(X, Y, Y, X) / (g(X, X) * g(Y, Y) - g(X, Y)^2)
where R is the Riemann curvature tensor, X and Y are any two vectors that span σ, and g is the Riemannian metric.
Properties
The sectional curvature has several important properties. For instance, it is symmetric in its arguments, meaning that if we switch the two vectors that define the plane, the sectional curvature remains the same. This is a direct consequence of the symmetry of the Riemann curvature tensor.
Another key property is that the sectional curvature is invariant under the action of the orthogonal group. This means that if we apply an orthogonal transformation to the plane, the sectional curvature does not change.
Interpretation
The sectional curvature can be interpreted in several ways. One of the most intuitive interpretations is in terms of the Gaussian curvature of the surface spanned by geodesics starting at the point p in the directions of the vectors that define the plane. In this sense, the sectional curvature measures how much the surface deviates from being flat.
Another interpretation is in terms of the deviation of geodesics from the Euclidean case. If the sectional curvature is positive, geodesics tend to converge, while if it is negative, they tend to diverge. In the case of zero sectional curvature, geodesics behave as in Euclidean space.
Examples
There are many examples of manifolds with constant sectional curvature. For instance, the Euclidean space has zero sectional curvature, the sphere has positive sectional curvature, and the hyperbolic space has negative sectional curvature.
Sectional Curvature in General Relativity
In the context of general relativity, the sectional curvature plays a crucial role. The Einstein field equations, which describe the dynamics of the gravitational field, can be written in terms of the Ricci curvature, which is a contraction of the Riemann curvature tensor. The sectional curvature, being a function of the Riemann curvature tensor, provides valuable information about the geometry of spacetime.