Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature <math>K(\sigma_p)</math> depends on a two-dimensional plane <math>\sigma_p</math> in the tangent space at p. It is the Gaussian curvature of that section — the surface which has the plane <math>\sigma_p</math> as a tangent plane at p, obtained from geodesics which start at p in the directions of <math>\sigma_p</math> (in other words, the image of <math>\sigma_p</math> under the exponential map at p). Formally, the sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.
The sectional curvature determines the curvature tensor completely and is a very useful geometric notion.
Definition
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, <math>u</math> and <math>v</math>, we can define
- <math>K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\cdot\langle v,v\rangle-\langle u,v\rangle^2}</math>
Here <math>R</math> is the Riemann curvature tensor.
It can be shown that <math>K(u,v)</math> depends only on the 2-plane <math>\sigma</math> spanned by <math>u</math> and <math>v</math>. It is called sectional curvature of the 2-plane <math>\sigma</math>.
Manifolds with constant sectional curvature
Riemannian manifolds with constant sectional curvature are the most simple. These are called space forms. By rescaling the metric there are three possible cases
- negative curvature −1, hyperbolic geometry
- zero curvature, Euclidean geometry
- positive curvature +1, elliptic geometry
The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit sphere. They are the only complete, simply connected Riemannian manifolds of given sectional curvature. All other complete constant curvature manifolds are quotients of those by some group of isometries.
Properties
- A complete Riemannian manifold has non-negative sectional curvature if and only if the function <math>f_p(x)=dist^2(p,x)</math> is 1-concave for all points p.
- A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function <math>f_p(x)=dist^2(p,x)</math> is 1-convex.
Manifolds with non-positive sectional curvature
Cartan showed that if M is a complete and simply connected manifold with non-positive sectional curvature, then it is diffeomorphic to a Euclidean space. Therefore the topological structure of a complete non-positively curved manifold is determined by its fundamental group.
Manifolds with positive sectional curvature
There is still known little about the structure of positively curved manifolds. It follows from the soul theorem that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively curved manifolds, there are two classical results:
- It follows from the Myers theorem that the fundamental group of such manifold is finite.
- It follows from the Synge theorem that the fundamental group of such manifold in even dimensions is 0, if orientable and <math>\Bbb Z_2</math> otherwise. In odd dimensions a positively curved manifold is always orientable.
Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures (e.g. the Hopf conjecture on whether there is a metric of positive sectional curvature on <math>\Bbb S^2\times\Bbb S^2</math>). The most typical way of constructing new examples is the following corollary from the O’Neill curvature formulas: if <math>(M,g)</math> is a Riemannian manifold admitting a free isometric action of a Lie group G, and M has positive sectional curvature on all 2-planes ortogonal to the orbits of G, then the manifold <math>M/G</math> with the quotient metric has positive sectional curvature. This fact allows to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples:
- The Berger spaces <math>B^7=SO(5)/SO(3)</math> and <math>B^{13}=SU(5)/Sp(2)\cdot\Bbb S^1</math>.
- The Wallach spaces (or the homogeneous flag manifolds): <math>W^6=SU(3)/T^2</math>, <math>W^{12}=Sp(3)/Sp(1)^3</math> and <math>W^{24}=F_4/Spin(8)</math>.
- The Aloff-Wallach spaces <math>W^7_{p,q}=SU(3)/diag(z^p,z^q,\overline{z}^{p+q})</math>.
- The Eschenburg spaces <math>E_{k,l}=diag(z^{k_1},z^{k_2},z^{k_3})\backslash SU(3)/diag(z^{l_1},z^{l_2},z^{l_3})^{-1}</math>.
- The Bazaikin spaces <math>B^{13}_p=diag(z_1^{p_1},\dots,z_1^{p_5})\backslash U(5)/diag(z_2A,1)^{-1}</math>, where <math>A\in Sp(2)\subset SU(4)</math>.
References
- J. Milnor, Morse Theory
External Links
- Wolfgang Ziller, Examples of manifolds with non-negative sectional curvature
See also
- Riemann curvature tensor
- curvature of Riemannian manifolds
- curvature
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