Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvaturenegatively curvednon-positive sectional curvaturenon-positively curvedsectional
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.wikipedia
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Space form

space forms
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K.

Hyperbolic space

hyperbolic 3-spacehyperbolic planehyperbolic 4-space
Hyperbolic n-space, denoted H n, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with a constant negative sectional curvature.

Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometry
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics 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 exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ p (in other words, the image of σ p under the exponential map at p).

Cartan–Hadamard theorem

Cartan-Hadamard theorem
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.
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
The sectional curvature determines the curvature tensor completely.
The Gaussian curvature coincides with the sectional curvature of the surface.

Comparison theorem

comparecomparison methodcomparison theorem in Riemannian geometry
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.

Toponogov's theorem

Toponogov theoremToponogov's triangle comparison theoremV. Toponogov
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. Toponogov's theorem affords a characterization of sectional curvature in terms of how "fat" geodesic triangles appear when compared to their Euclidean counterparts.
Let M be an m-dimensional Riemannian manifold with sectional curvature K satisfying

Soul theorem

soul conjecturesouls
The soul theorem implies that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold.
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case.

Preissman's theorem

Preissman's theorem restricts the fundamental group of negatively curved compact manifolds.
In Riemannian geometry, a field of mathematics, Preissman's theorem is a statement that restricts the possible topology of a negatively curved compact Riemannian manifold M.

Myers's theorem

Myers theoremMyers' theoremBonnet–Myers theorem
A weaker result, due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by k.

Curvature

curvednegative curvatureextrinsic curvature
The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of tidal force (this is one way of thinking of the sectional curvature).

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

Tangent space

tangent planetangenttangent vector
The sectional curvature K(σ p ) depends on a two-dimensional plane σ p in the tangent space at a point p of the manifold.

Surface (topology)

surfaceclosed surfacesurfaces
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ p (in other words, the image of σ p under the exponential map at p).

Geodesic

geodesicsgeodesic flowgeodesic equation
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ p (in other words, the image of σ p under the exponential map at p).

Grassmannian

GrassmanniansGrassmann manifold Grassmannian manifolds
The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

Fiber bundle

structure grouplocal trivializationtrivial bundle
The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

Riemannian manifold

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

Linear independence

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

Tangent vector

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

Orthonormality

orthonormalorthonormal setorthonormal sequence
In particular, if u and v are orthonormal, then

Euclidean space

EuclideanspaceEuclidean vector space
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.