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 linear subspace σ_{p} of the tangent space at a point *p* of the manifold. It can be defined geometrically as 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 sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

- Definition
- Alternative definitions
- Manifolds with constant sectional curvature
- The model examples
- Scaling
- Toponogov's theorem
- Manifolds with non-positive sectional curvature
- Manifolds with positive sectional curvature
- Manifolds with non-negative sectional curvature
- See also
- References

The sectional curvature determines the curvature tensor completely.

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

Here *R* is the Riemann curvature tensor, defined here by the convention Some sources use the opposite convention in which case *K(u,v)* must be defined with in the numerator instead of ^{ [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 are linearly independent and span the same two-dimensional linear subspace of the tangent space as , then 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. Let be a two-dimensional plane in . Let for sufficiently small denote the image under the exponential map at of the unit circle in , and let denote the length of . Then it can be proven that

as , for some number . This number at is the sectional curvature of at .^{ [2] }

One says that a Riemannian manifold has "constant curvature " if for all two-dimensional linear subspaces and for all

The Schur lemma states that if *(M,g)* is a connected Riemannian manifold with dimension at least three, and if there is a function such that for all two-dimensional linear subspaces and for all then *f* must be constant and hence *(M,g)* has constant curvature.

A Riemannian manifold with constant sectional curvature is called a space form. If denotes the constant value of the sectional curvature, then the curvature tensor can be written as

for any

Proof |

Briefly: one polarization argument gives a formula for a second (equivalent) polarization argument gives a formula for and a combination with the first Bianchi identity recovers the given formula for From the definition of sectional curvature, we know that whenever are linearly independent, and this easily extends to the case that are linearly dependent since both sides are then zero. Now, given arbitrary Secondly, by multilinearity, it equals which, recalling the Riemannian symmetry can be simplified to Setting these two computations equal to each other and canceling terms, one finds Since for any Secondly, by multilinearity, it equals which by the new formula equals Setting these two computations equal to each other shows Swap and , then add this to the Bianchi identity to get Subtract these two equations, making use of the symmetry to get |

Since any Riemannian metric is parallel with respect to its Levi-Civita connection, this shows that the Riemann tensor of any constant-curvature space is also parallel. The Ricci tensor is then given by and the scalar curvature is In particular, any constant-curvature space is Einstein and has constant scalar curvature.

Given a positive number define

- to be the standard Riemannian structure
- to be the sphere with given by the pullback of the standard Riemannian structure on by the inclusion map
- to be the ball with

In the usual terminology, these Riemannian manifolds are referred to as Euclidean space, the n-sphere, and hyperbolic space. Here, the point is that each is a complete connected smooth Riemannian manifold with constant curvature. To be precise, the Riemannian metric has constant curvature 0, the Riemannian metric has constant curvature and the Riemannian metric has constant curvature

Furthermore, these are the 'universal' examples in the sense that if 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 according to the constant curvatures of the above examples.

If 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 with the pullback Riemannian metric Since is, by topological principles, a covering map, the Riemannian manifold is locally isometric to , and so it is a smooth, connected, and simply-connected complete Riemannian manifold with the same constant curvature as It must then be isometric one of the above model examples. Note that the deck transformations of the universal cover are isometries relative to the metric

The study of Riemannian manifolds with constant negative curvature, called hyperbolic geometry, is particularly noteworthy as it exhibits many noteworthy phenomena.

Let be a smooth manifold, and let be a positive number. Consider the Riemannian manifold The curvature tensor, as a multilinear map is unchanged by this modification. Let be linearly independent vectors in . Then

So multiplication of the metric by multiplies all of the sectional curvatures by

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 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

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. In particular, it is aspherical: the homotopy groups 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. 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.

Little is known about the structure of positively curved 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:

- It follows from the Myers theorem that the fundamental group of such a manifold is finite.
- It follows from the Synge theorem that the fundamental group of such a manifold in even dimensions is 0, if orientable and 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 ). 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 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 ( Ziller 2007 ):

- The Berger spaces and .
- The Wallach spaces (or the homogeneous flag manifolds): , and .
- The Aloff–Wallach spaces .
- The Eschenburg spaces
- The Bazaikin spaces , where .

Cheeger and Gromoll proved their soul theorem which states that any non-negatively curved complete non-compact manifold has a totally convex compact submanifold such that is diffeomorphic to the normal bundle of . Such an is called the soul of . In particular, this theorem implies that is homotopic to its soul which has the dimension less than .

In the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is *flat*, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In the mathematical field of Riemannian geometry, the **scalar curvature** is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.

In mathematics, and especially differential geometry and gauge theory, a **connection** on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In mathematics, the **covariant derivative** is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry.

In differential geometry, the **second fundamental form** is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

In differential geometry, the **Laplace–Beltrami operator** is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the **Ricci decomposition** is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry.

In mathematics, the **Fubini–Study metric** is a Kähler metric on projective Hilbert space, that is, on a complex projective space **CP**^{n} endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

In mathematics, a **metric connection** is a connection in a vector bundle *E* equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to:

In 3-dimensional topology, a part of the mathematical field of geometric topology, the **Casson invariant** is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.

In Riemannian geometry and pseudo-Riemannian geometry, the **Gauss–Codazzi equations** are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of a Riemannian or pseudo-Riemannian manifold.

The **Yamabe problem** refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:

Let (

M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metricfghas constant scalar curvature.

In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite.

In physics and mathematics, and especially differential geometry and gauge theory, the **Yang–Mills equations** are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the **Yang–Mills action functional**. They have also found significant use in mathematics.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.

In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 : in particular, it is a CAT(-1/4) space.

- ↑ Gallot, Hulin & Lafontaine 2004, Section 3.A.2.
- ↑ Gallot, Hulin & Lafontaine 2004, Section 3.D.4.

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*Annals of Mathematics*, Second Series, Annals of Mathematics,**96**(3): 413–443, doi:10.2307/1970819, JSTOR 1970819, MR 0309010 . - Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004).
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*Annals of Mathematics*, Second Series, Annals of Mathematics,**90**(1): 75–90, doi:10.2307/1970682, JSTOR 1970682, MR 0247590 . - Milnor, J. (1963).
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