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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''(σ''p'') depends on a two-dimensional plane σ''p'' in the tangent space at ''p''. It is the [[Gaussian curvature]] of the [[surface]] which has the plane σ''p'' as a tangent plane at ''p'', obtained from [[geodesic]]s which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the [[exponential map]] at ''p''). The sectional curvature is a smooth real-valued function on the 2-[[Grassmannian]] [[fiber bundle|bundle]] over the manifold.

The sectional curvature determines the [[Riemann curvature tensor|curvature tensor]] completely.

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

:<math>K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2}</math>

Here ''R'' is the [[Riemann curvature tensor]].

In particular, if ''u'' and ''v'' are [[orthonormal]], then
:<math>K(u,v) = \langle R(u,v)v,u\rangle.</math>
The sectional curvature in fact depends only on the 2-plane σ''p'' in the tangent space at ''p'' spanned by ''u'' and ''v''. It is called the '''sectional curvature of the 2-plane σ''p''''', and is denoted ''K''(σ''p'').

==Manifolds with constant sectional curvature==

[[Riemannian manifold]]s with constant sectional curvature are the most simple. These are called [[space form]]s. 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 [[n-sphere|sphere]]. They are the only [[Complete space|complete]], [[simply connected]] Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of [[isometry|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.

==Toponogov's theorem==
[[Toponogov's theorem]] affords a characterization of sectional curvature in terms of how "fat" geodesic triangles appear when compared to their Euclidean counterparts. The basic intuition is that, if a space is positively curved, then the edge of a triangle opposite some given vertex will tend to bend away from that vertex, whereas if a space is negatively curved, then the opposite edge of the triangle will tend to bend towards the vertex.

More precisely, let ''M'' be a [[complete space|complete]] Riemannian manifold, and let ''xyz'' be a geodesic triangle in ''M'' (a triangle each of whose sides is a length-minimizing geodesic). Finally, let ''m'' be the midpoint of the geodesic ''xy''. If ''M'' has non-negative curvature, then for all sufficiently small triangles
:<math>d(z,m)^2\ge \tfrac12d(z,x)^2 + \tfrac12d(z,y)^2 - \tfrac14d(x,y)^2</math>
where ''d'' is the [[distance function]] on ''M''. The case of equality holds precisely when the curvature of ''M'' vanishes, and the right-hand side represents the distance from a vertex to the opposite side of a geodesic triangle in Euclidean space having the same side-lengths as the triangle ''xyz''. This makes precise the sense in which triangles are "fatter" in positively curved spaces. In non-positively curved spaces, the inequality goes the other way:
:<math>d(z,m)^2\le \tfrac12d(z,x)^2 + \tfrac12d(z,y)^2 - \tfrac14d(x,y)^2.</math>

If tighter bounds on the sectional curvature are known, then this property generalizes to give a [[comparison theorem]] between geodesic triangles in ''M'' and those in a suitable simply connected space form; see [[Toponogov's theorem]]. Simple consequences of the version stated here are:

*A complete Riemannian manifold has non-negative sectional curvature if and only if the function <math>f_p(x)=\operatorname{dist}^2(p,x)</math> is 1-[[Glossary of Riemannian and metric geometry|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)=\operatorname{dist}^2(p,x)</math> is 1-[[Glossary of Riemannian and metric geometry|convex]].

==Manifolds with non-positive sectional curvature==
In 1928, [[Élie Cartan]] proved the [[Cartan–Hadamard theorem]]: if ''M'' is a [[Complete space|complete]] manifold with non-positive sectional curvature, then its [[universal cover]] is [[diffeomorphic]] to a [[Euclidean space]]. In particular, it is [[Aspherical space|aspherical]]: the [[homotopy groups]] <math>\pi_i(M)</math> for ''i'' ≥ 2 are trivial. Therefore, the topological structure of a complete non-positively curved manifold is determined by its [[fundamental group]]. [[Preissman's theorem]] restricts the fundamental group of negatively curved compact manifolds.

==Manifolds with positive sectional curvature==
Little is known about the structure of positively curved manifolds. The [[soul theorem]] ({{harvnb|Cheeger|Gromoll|1972}}; {{harvnb|Gromoll|Meyer|1969}}) implies 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 orthogonal to the orbits of G, then the manifold <math>M/G</math> with the quotient metric has positive sectional curvature. This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples {{harv|Ziller|2007}}:

*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)/\operatorname{diag}(z^p,z^q,\overline{z}^{p+q})</math>.

*The Eschenburg spaces <math>E_{k,l}=\operatorname{diag}(z^{k_1},z^{k_2},z^{k_3})\backslash SU(3)/\operatorname{diag}(z^{l_1},z^{l_2},z^{l_3})^{-1}.</math>

*The Bazaikin spaces <math>B^{13}_p=\operatorname{diag}(z_1^{p_1},\dots,z_1^{p_5})\backslash U(5)/\operatorname{diag}(z_2A,1)^{-1}</math>, where <math>A\in Sp(2)\subset SU(4)</math>.

==References==
*{{Citation | doi=10.2307/1970819 | last1=Cheeger | first1=Jeff | last2=Gromoll | first2=Detlef | title=On the structure of complete manifolds of nonnegative curvature | mr=0309010 | year=1972 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=96 | pages=413–443 | issue=3 | publisher=Annals of Mathematics | jstor=1970819}}.
*{{Citation | doi=10.2307/1970682 | last1=Gromoll | first1=Detlef | last2=Meyer | first2=Wolfgang | title=On complete open manifolds of positive curvature | mr=0247590 | year=1969 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=90 | pages=75–90 | issue=1 | publisher=Annals of Mathematics | jstor=1970682}}.
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Morse theory | publisher=[[Princeton University Press]] | series=Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 | mr=0163331 | year=1963}}.
* {{Citation | last1=Petersen | first1=Peter | title=Riemannian geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-29246-5 | mr=2243772 | year=2006 | volume=171}}.
*{{cite arxiv|first=Wolfgang|last=Ziller|title=Examples of manifolds with non-negative sectional curvature|eprint=math/0701389|year=2007}}.

==See also==

*[[Riemann curvature tensor]]
*[[curvature of Riemannian manifolds]]
*[[curvature]]

{{curvature}}

[[Category:Riemannian geometry]]
[[Category:Curvature (mathematics)]]

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