The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.
[1] Note that the linear independence of u and v forces the denominator in the above expression to be nonzero, so that K(u,v) is well-defined.
In particular, if u and v are orthonormal, then the definition takes on the simple form It is straightforward to check that if
So one may consider the sectional curvature as a real-valued function whose input is a two-dimensional linear subspace of a tangent space.
Alternatively, the sectional curvature can be characterized by the circumference of small circles.
The Schur lemma states that if (M,g) is a connected Riemannian manifold with dimension at least three, and if there is a function
A Riemannian manifold with constant sectional curvature is called a space form.
can be simplified to Setting these two computations equal to each other and canceling terms, one finds Since w is arbitrary this shows that for any u,v.
In particular, any constant-curvature space is Einstein and has constant scalar curvature.
Here, the point is that each is a complete connected smooth Riemannian manifold with constant curvature.
is a smooth, connected, and simply-connected complete Riemannian manifold with constant curvature, then it is isometric to one of the above examples; the particular example is dictated by the value of the constant curvature of
is a smooth and connected complete Riemannian manifold with constant curvature, but is not assumed to be simply-connected, then consider the universal covering space
is, by topological principles, a covering map, the Riemannian manifold
, and so it is a smooth, connected, and simply-connected complete Riemannian manifold with the same constant curvature as
Note that the deck transformations of the universal cover are isometries relative to the metric
The study of Riemannian manifolds with constant negative curvature is called hyperbolic geometry.
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.
If M has non-negative curvature, then for all sufficiently small triangles 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: 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: In 1928, Élie Cartan proved the Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space.
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.
The Cartan–Hadamard conjecture states that the classical isoperimetric inequality should hold in all simply connected spaces of non-positive curvature, which are called Cartan-Hadamard manifolds.
The soul theorem (Cheeger & Gromoll 1972; 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: 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
The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if
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
This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples (Ziller 2007): Cheeger and Gromoll proved their soul theorem which states that any non-negatively curved complete non-compact manifold