Sectional curvature

The sectional curvature is a quantity of Riemannian geometry, a branch of mathematics. With their help, you can describe the curvature of a -dimensional Riemannian manifold. Here, each ( two-dimensional ) plane is assigned a number in the tangent space at a point of this manifold as curvature. The interface curvature can be understood as a generalization of the Gaussian curvature. The name comes from the fact that you speak defines a section through the manifold in the direction of the given plane and determines the Gaussian curvature of the resulting surface.

Definition

Given a Riemannian manifold, and a point in a two-dimensional subspace ( plane) of the tangent space of the point. Let and be two tangent vectors that span this plane. with

Is the area of ​​the parallelogram spanned by and called, called the Riemannian curvature tensor.

Then, the size depends

Only of the plane from, but not on the choice of them spanning vectors and. It therefore also writes for and calls this the sectional curvature of.

Since different sign conventions for the Riemann curvature tensor exist, the sectional curvature is depending on the context by

Defined. In this article, however, the first convention.

Relationship to the Gaussian curvature

Let be a 2- dimensional submanifold of the Euclidean space and the induced metric on. For each point and each base of the Schnittkrümung

Equal to the Gaussian curvature of the point. That one, the Gaussian curvature can portray it is a consequence of Gauss's Theorema egregium.

Relations with other curvature quantities

• All information provided by the Riemann curvature tensor, are included in the sectional curvature. So you can recover from the sectional curvature of the Riemannian curvature tensor. Be namely and two tensors which satisfy the symmetry properties, and the Bianchi identity. Then every pair of linearly independent vectors, the equation as follows.
• As you can recover the Riemannian curvature tensor of the sectional curvature, one can find a relation between the Ricci curvature and the sectional curvature. For this purpose let be an orthonormal basis of the tangent space as valid The Ricci curvature is completely determined by the formula, since the Ricci tensor is symmetric. Has the underlying Riemannian manifold of dimension constant sectional curvature, the simplified formula is
• For the scalar curvature we obtain the similar formula which again is an orthonormal basis of the tangent space. If the sectional curvature is constant, applies

Examples

• The sectional curvature of the Euclidean space is constant is zero because of the Riemann curvature tensor is defined such that it vanishes for all points from.
• The sphere with radius sectional curvature. Since this is isotropic and homogeneous, the sectional curvature is constant and it is sufficient to determine this at the North Pole. With the exponential at the North Pole is called. Furthermore, it is the two -dimensional subspace of the tangent space, which is spanned by. Now is a manifold which is isometric to. From this it is known that the amounts Gaußkrümmung. Therefore, the - dimensional sphere has the sectional curvature.
• The hyperbolic space has sectional curvature

Applications

Manifolds with constant curvature

As in other areas of mathematics, one tries to classify objects in the Riemannian geometry. In Riemannian geometry, the corresponding Riemannian manifolds are classified. Thus one sees two manifolds considered equal if there is an isometric mapping between them. The sectional curvature, since it depends on Riemannian metric is an important invariant of Riemannian manifolds. When complete, simply connected Riemannian manifolds with constant sectional curvature classification is relatively easy because there are only three cases to consider. If the Riemannian manifold the dimension and the constant positive sectional curvature, then it is isometric (equal) to the -dimensional sphere with radius. If the sectional curvature constant zero it is called the multiplicity of flat and is isometric to the Euclidean space and in the case that the manifold has negative sectional curvature, so it corresponds to the -dimensional hyperbolic space.

One does not now consider only the simply connected manifolds, but all complete and coherent manifolds with constant sectional curvature, so their classification is more complicated. The fundamental group of manifolds no longer disappears. It can now show that such manifolds are isometric to. Where one of the three rooms of the above section is therefore for or and a discrete subgroup of the isometry group of is, which operates freely and properly discontinuously on. This group is isomorphic to the fundamental group of.

Manifolds with negative curvature

Élie Cartan generalized a result in 1928 by Jacques Hadamard, which states in modern formulation that the exponential map is a universal covering for non- positive sectional curvature. This statement is now called the set of Cartan -Hadamard. There are different formulations of the sentence. The version for Riemannian manifolds is accurate:

This set is remarkable, among other reasons, because it provides a connection between a local variable and a global size of a differentiable manifold. Such statements are also local-global theorems mentioned. In this case, the average curvature of the manifold is the local size, because the curvature is defined for each section. Under the assumption that the manifold is simply connected, it is diffeomorphic to the set according to what is a global, differentialtopologische property that has nothing to do with the Riemannian metric. From the theorem now follows that compact, complete, simply connected manifolds such as the sphere is, however, always somewhere positive sectional curvature. For, because the sphere is compact, it can not diffeomorphic to be. From the condition of non-positive average curvature is so strong restrictions receives the topology, which can support the manifold. With tools of algebraic topology can be shown that the homotopy groups of manifolds which satisfy the conditions of the theorem vanish for.

Manifolds with positive curvature

One result from the field manifolds with positive sectional curvature is the set of Bonnet. This local-global theorem brings the sectional curvature with the topological properties of compactness and finite fundamental group in conjunction. Precise says the sentence: