# Sectional curvature

Last updated

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 2. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp 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 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). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

## Contents

The sectional curvature determines the curvature tensor completely.

## Definition

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

$K(u,v)={\langle R(u,v)v,u\rangle \over \langle u,u\rangle \langle v,v\rangle -\langle u,v\rangle ^{2}}$ Here R is the Riemann curvature tensor, defined here by the convention $R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w.$ Some sources use the opposite convention $R(u,v)w=\nabla _{v}\nabla _{u}w-\nabla _{u}\nabla _{v}w-\nabla _{[v,u]}w,$ in which case K(u,v) must be defined with $\langle R(u,v)u,v\rangle$ in the numerator instead of $\langle R(u,v)v,u\rangle .$ 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

$K(u,v)=\langle R(u,v)v,u\rangle .$ It is straightforward to check that if $u,v\in T_{p}M$ are linearly independent and span the same two-dimensional linear subspace of $T_{p}M$ as $x,y\in T_{p}M$ , then $K(u,v)=K(x,y).$ So one may consider the sectional curvature as a real-valued function whose input is a two-dimensional linear subspace of a tangent space.

## Manifolds with constant sectional curvature

One says that a Riemannian manifold has "constant curvature $\kappa$ " if $\operatorname {sec} (P)=\kappa$ for all two-dimensional linear subspaces $P\subset T_{p}M$ and for all $p\in M.$ The Schur lemma states that if (M,g) is a connected Riemannian manifold with dimension at least three, and if there is a function $f:M\to \mathbb {R}$ such that $\operatorname {sec} (P)=f(p)$ for all two-dimensional linear subspaces $P\subset T_{p}M$ and for all $p\in M,$ 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 $\kappa$ denotes the constant value of the sectional curvature, then the curvature tensor can be written as

$R(u,v)w=\kappa {\big (}\langle v,w\rangle u-\langle u,w\rangle v{\big )}$ for any $u,v,w\in T_{p}M.$ 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 $\operatorname {Ric} =(n-1)\kappa g$ and the scalar curvature is $n(n-1)\kappa .$ In particular, any constant-curvature space is Einstein and has constant scalar curvature.

### The model examples

Given a positive number $a,$ define

• $\left(\mathbb {R} ^{n},g_{\mathbb {R} ^{n}}\right)$ to be the standard Riemannian structure
• $\left(S^{n}(a),g_{S^{n}(a)}\right)$ to be the sphere $S^{n}(a)\equiv \left\{x\in \mathbb {R} ^{n+1}:|x|=a\right\}$ with $g_{S^{n}(a)}$ given by the pullback of the standard Riemannian structure on $\mathbb {R} ^{n+1}$ by the inclusion map $S^{n}(a)\to \mathbb {R} ^{n+1}$ • $\left(H^{n}(a),g_{H^{n}(a)}\right)$ to be the ball $H^{n}(a)\equiv \left\{x\in \mathbb {R} ^{n}:|x| with $g_{H^{n}(a)}=a^{2}{\frac {\left(a^{2}-|x|^{2}\right)\left(dx_{1}^{2}+\cdots +dx_{n}^{2}\right)-\left(x_{1}\,dx_{1}+\cdots +x_{n}\,dx_{n}\right)^{2}}{\left(a^{2}-|x|^{2}\right)^{2}}}.$ 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 $g_{\mathbb {R} ^{n}}$ has constant curvature 0, the Riemannian metric $g_{S^{n}(a)}$ has constant curvature $a^{-2},$ and the Riemannian metric $g_{H^{n}(a)}$ has constant curvature $-a^{-2}.$ Furthermore, these are the 'universal' examples in the sense that if $(M,g)$ 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 $g,$ according to the constant curvatures of the above examples.

If $(M,g)$ 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 $\p$ :{\widetilde {M}}\to M} with the pullback Riemannian metric $\pi ^{\ast }g.$ Since $\pi$ is, by topological principles, a covering map, the Riemannian manifold $({\widetilde {M}},\pi ^{\ast }g)$ is locally isometric to $(M,g)$ , and so it is a smooth, connected, and simply-connected complete Riemannian manifold with the same constant curvature as $g.$ 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 $\pi ^{\ast }g.$ The study of Riemannian manifolds with constant negative curvature, called hyperbolic geometry, is particularly noteworthy as it exhibits many noteworthy phenomena.

## Scaling

Let $(M,g)$ be a smooth manifold, and let $\lambda$ be a positive number. Consider the Riemannian manifold $(M,\lambda g).$ The curvature tensor, as a multilinear map $T_{p}M\times T_{p}M\times T_{p}M\to T_{p}M,$ is unchanged by this modification. Let $v,w$ be linearly independent vectors in $T_{p}M$ . Then

$K_{\lambda g}(v,w)={\frac {\lambda g\left(R^{\lambda g}(v,w)w,v\right)}{|v|_{\lambda g}^{2}|w|_{\lambda g}^{2}-\langle v,w\rangle _{\lambda g}^{2}}}={\frac {1}{\lambda }}{\frac {g\left(R^{g}(v,w)w,v\right)}{|v|_{g}^{2}|w|_{g}^{2}-\langle v,w\rangle _{g}^{2}}}={\frac {1}{\lambda }}K_{g}(v,w).$ So multiplication of the metric by $\lambda$ multiplies all of the sectional curvatures by $\lambda ^{-1}.$ ## 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 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

$d(z,m)^{2}\geq {\frac {1}{2}}d(z,x)^{2}+{\frac {1}{2}}d(z,y)^{2}-{\frac {1}{4}}d(x,y)^{2}$ 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:

$d(z,m)^{2}\leq {\frac {1}{2}}d(z,x)^{2}+{\frac {1}{2}}d(z,y)^{2}-{\frac {1}{4}}d(x,y)^{2}.$ 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 $f_{p}(x)=\operatorname {dist} ^{2}(p,x)$ is 1-concave for all points p.
• A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function $f_{p}(x)=\operatorname {dist} ^{2}(p,x)$ is 1-convex.

## Manifolds with non-positive sectional curvature

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 $\pi _{i}(M)$ 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.

## Manifolds with positive sectional curvature

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 $\mathbb {Z} _{2}$ 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 $\mathbb {S} ^{2}\times \mathbb {S} ^{2}$ ). 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 orthogonal to the orbits of G, then the manifold $M/G$ 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 $B^{7}=SO(5)/SO(3)$ and $B^{13}=SU(5)/\operatorname {Sp} (2)\cdot \mathbb {S} ^{1}$ .
• The Wallach spaces (or the homogeneous flag manifolds): $W^{6}=SU(3)/T^{2}$ , $W^{12}=\operatorname {Sp} (3)/\operatorname {Sp} (1)^{3}$ and $W^{24}=F_{4}/\operatorname {Spin} (8)$ .
• The Aloff–Wallach spaces $W_{p,q}^{7}=SU(3)/\operatorname {diag} \left(z^{p},z^{q},{\overline {z}}^{p+q}\right)$ .
• The Eschenburg spaces $E_{k,l}=\operatorname {diag} \left(z^{k_{1}},z^{k_{2}},z^{k_{3}}\right)\backslash SU(3)/\operatorname {diag} \left(z^{l_{1}},z^{l_{2}},z^{l_{3}}\right)^{-1}.$ • The Bazaikin spaces $B_{p}^{13}=\operatorname {diag} \left(z_{1}^{p_{1}},\dots ,z_{1}^{p_{5}}\right)\backslash U(5)/\operatorname {diag} (z_{2}A,1)^{-1}$ , where $A\in \operatorname {Sp} (2)\subset SU(4)$ .

## Manifolds with non-negative sectional curvature

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

## Related Research Articles

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality in many mathematical fields, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics. 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, that measures the extent to which the metric tensor is not locally isometric to that of 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, 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 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 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 differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. 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 Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.

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 CPn 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: Corner detection is an approach used within computer vision systems to extract certain kinds of features and infer the contents of an image. Corner detection is frequently used in motion detection, image registration, video tracking, image mosaicing, panorama stitching, 3D reconstruction and object recognition. Corner detection overlaps with the topic of interest point detection.

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 metric fg has constant scalar curvature.

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. The Yang–Mills equations arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. However, the Yang–Mills equations have independently found significant use within mathematics.

In the mathematical field of differential geometry, a Codazzi tensor is a symmetric 2-tensor whose covariant derivative is also symmetric. Such tensors arise naturally in the study of Riemannian manifolds with harmonic curvature or harmonic Weyl tensor. In fact, existence of Codazzi tensors impose strict conditions on the curvature tensor of the manifold. Also, the second fundamental form of an immersed hypersurface in a space form is a Codazzi tensor.

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 physical model of some natural phenomenon.

• Cheeger, Jeff; Gromoll, Detlef (1972), "On the structure of complete manifolds of nonnegative curvature", Annals of Mathematics , Second Series, Annals of Mathematics, 96 (3): 413–443, doi:10.2307/1970819, JSTOR   1970819, MR   0309010 .
• Gromoll, Detlef; Meyer, Wolfgang (1969), "On complete open manifolds of positive curvature", Annals of Mathematics , Second Series, Annals of Mathematics, 90 (1): 75–90, doi:10.2307/1970682, JSTOR   1970682, MR   0247590 .
• Milnor, John Willard (1963), Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, MR   0163331 .
• Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, 171 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN   978-0-387-29246-5, MR   2243772 .
• Ziller, Wolfgang (2007). "Examples of manifolds with non-negative sectional curvature". arXiv:..