Sectional curvature

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In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional plane σp in the tangent space at p. It is the Gaussian curvature of that section — the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp 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.

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[edit] Definition

Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define

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}

Here R is the Riemann curvature tensor.

It can be shown that K(u,v) depends only on the 2-plane σ spanned by u and v. It is called sectional curvature of the 2-plane σ.

[edit] 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

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.

If for each point in a connected Riemannian manifold (of dimension three or greater) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.

[edit] Properties

  • A complete Riemannian manifold has non-negative sectional curvature if and only if the function fp(x) = dist2(p,x) is 1-concave for all points p.
  • A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function fp(x) = dist2(p,x) is 1-convex.

[edit] Manifolds with non-positive sectional curvature

Cartan showed that if M is a complete manifold with non-positive sectional curvature, then its universal cover M' is diffeomorphic to a Euclidean space. In particular, the homotopy groups πi(M) for i \geq 2 are trivial. Therefore, the topological structure of a complete non-positively curved manifold is determined by its fundamental group.

[edit] 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 \Bbb Z_2 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 \Bbb S^2\times\Bbb S^2). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if (M,g) 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 M / G 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 B7 = SO(5) / SO(3) and B^{13}=SU(5)/Sp(2)\cdot\Bbb S^1.
  • The Wallach spaces (or the homogeneous flag manifolds): W6 = SU(3) / T2, W12 = Sp(3) / Sp(1)3 and W24 = F4 / Spin(8).
  • The Aloff-Wallach spaces W^7_{p,q}=SU(3)/diag(z^p,z^q,\overline{z}^{p+q}).
  • The Eschenburg spaces 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}.
  • The Bazaikin spaces B^{13}_p=diag(z_1^{p_1},\dots,z_1^{p_5})\backslash U(5)/diag(z_2A,1)^{-1}, where A\in Sp(2)\subset SU(4).

[edit] References

[edit] External links

[edit] See also

v • d • e
Different notions of curvature defined in differential geometry
Differential geometry of curves
Differential geometry of surfaces
Riemannian geometry
Curvature of connections
Languages