Glossary of Riemannian and metric geometry

"Radius of convexity" redirects here. For the anatomical feature of the radius bone, see Convexity of radius.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.

• Connection
• Curvature
• Metric space
• Riemannian manifold

See also:

• Glossary of general topology
• Glossary of differential geometry and topology
• List of differential geometry topics

Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or $|xy{|}_X$ denotes the distance between points x and y in X. Italic word denotes a self-reference to this glossary. A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general mathematical usage.

A

Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2) Almost flat manifold Arc-wise isometry the same as path isometry. Autoparallel the same as totally geodesic.

B

Barycenter, see center of mass. bi-Lipschitz map. A map $f:X\to Y$ is called bi-Lipschitz if there are positive constants c and C such that for any x and y in X

$c|xy{|}_X\le |f(x)f(y){|}_Y\le C|xy{|}_X$

Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by

$B_{\gamma }(p)=\underset{t\to \mathrm{\infty }}{\lim }(|\gamma (t)-p|-t)$

C

Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to Rⁿ via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of Alexandrov is a (globally) CAT(0) space. Cartan extended Einstein's General relativity to Einstein–Cartan theory, using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides affine torsion, which allows for non-symmetric curvature tensors and the incorporation of spin–orbit coupling. Center of mass. A point $q\in M$ is called the center of mass of the points $p_1,p_2,\dots ,p_k$ if it is a point of global minimum of the function

$f(x)=\sum _i|p_{i}x{|}^2.$

Such a point is unique if all distances $|p_i{p}_j|$ are less than the convexity radius. Christoffel symbol Collapsing manifold Complete manifold Complete metric space Completion Conformal map is a map which preserves angles. Conformally flat a manifold M is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat. Conjugate points two points p and q on a geodesic $\gamma$ are called conjugate if there is a Jacobi field on $\gamma$ which has a zero at p and q. Convex function. A function f on a Riemannian manifold is a convex if for any geodesic $\gamma$ the function $f\circ \gamma$ is convex. A function f is called $\lambda$-convex if for any geodesic $\gamma$ with natural parameter $t$, the function $f\circ \gamma (t)-\lambda t^2$ is convex. Convex A subset K of a Riemannian manifold M is called convex if for any two points in K there is a unique shortest path connecting them which lies entirely in K, see also totally convex. Convexity radius at a point $p$ of a Riemannian manifold is the supremum of radii of balls centered at $p$ that are (totally) convex. The convexity radius of the manifold is the infimum of the convexity radii at its points; for a compact manifold this is a positive number. Sometimes the additional requirement is made that the distance function to $p$ in these balls is convex. Cotangent bundle Covariant derivative Cut locus

D

Diameter of a metric space is the supremum of distances between pairs of points. Developable surface is a surface isometric to the plane. Dilation same as Lipschitz constant

E

Exponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)

F

Finsler metric First fundamental form for an embedding or immersion is the pullback of the metric tensor. Flat manifold

G

Geodesic is a curve which locally minimizes distance. Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories are of the form $(\gamma (t),{\gamma }^{\prime }(t))$ where $\gamma$ is a geodesic. Gromov-Hausdorff convergence Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.

H

Hadamard space is a complete simply connected space with nonpositive curvature. Horosphere a level set of Busemann function.

I

Injectivity radius The injectivity radius at a point p of a Riemannian manifold is the supremum of radii for which the exponential map at p is a diffeomorphism. The injectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points. See also cut locus. For complete manifolds, if the injectivity radius at p is a finite number r, then either there is a geodesic of length 2r which starts and ends at p or there is a point q conjugate to p (see conjugate point above) and on the distance r from p. For a closed Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic or the minimal distance between conjugate points on a geodesic. Infranilmanifold Given a simply connected nilpotent Lie group N acting on itself by left multiplication and a finite group of automorphisms F of N one can define an action of the semidirect product $N⋊F$ on N. An orbit space of N by a discrete subgroup of $N⋊F$ which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold. Isometry is a map which preserves distances. Intrinsic metric

J

Jacobi field A Jacobi field is a vector field on a geodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics ${\gamma }_{\tau }$ with ${\gamma }_0=\gamma$, then the Jacobi field is described by

$J(t)={\frac{\partial {\gamma }_{\tau }(t)}{\partial \tau }|}_{\tau =0}.$

Jordan curve

K

Kähler-Einstein metric Kähler metric Killing vector field

L

Length metric the same as intrinsic metric. Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds. Lipschitz constant of a map is the infimum of numbers L such that the given map is L-Lipschitz. Lipschitz convergence the convergence of metric spaces defined by Lipschitz distance. Lipschitz distance between metric spaces is the infimum of numbers r such that there is a bijective bi-Lipschitz map between these spaces with constants exp(-r), exp(r). Lipschitz map Logarithmic map, or logarithm, is a right inverse of Exponential map.

M

Mean curvature Metric ball Metric tensor Minimal surface is a submanifold with (vector of) mean curvature zero.

N

Natural parametrization is the parametrization by length. Net. A subset S of a metric space X is called $ϵ$-net if for any point in X there is a point in S on the distance $\le ϵ$. This is distinct from topological nets which generalize limits. Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented $S^1$-bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice. Normal bundle: associated to an embedding of a manifold M into an ambient Euclidean space ${\mathbb{ℝ}}^N$, the normal bundle is a vector bundle whose fiber at each point p is the orthogonal complement (in ${\mathbb{ℝ}}^N$) of the tangent space $T_{p}M$. Nonexpanding map same as short map.

P

Parallel transport Path isometry Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space. Principal curvature is the maximum and minimum normal curvatures at a point on a surface. Principal direction is the direction of the principal curvatures. Proper metric space is a metric space in which every closed ball is compact. Equivalently, if every closed bounded subset is compact. Every proper metric space is complete. Pseudo-Riemannian manifold

Q

Quasigeodesic has two meanings; here we give the most common. A map $f:I\to Y$ (where $I\subseteq \mathbb{ℝ}$ is a subinterval) is called a quasigeodesic if there are constants $K\ge 1$ and $C\ge 0$ such that for every $x,y\in I$

$\frac{1}{K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.$

Note that a quasigeodesic is not necessarily a continuous curve. Quasi-isometry. A map $f:X\to Y$ is called a quasi-isometry if there are constants $K\ge 1$ and $C\ge 0$ such that

$\frac{1}{K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.$

and every point in Y has distance at most C from some point of f(X). Note that a quasi-isometry is not assumed to be continuous. For example, any map between compact metric spaces is a quasi isometry. If there exists a quasi-isometry from X to Y, then X and Y are said to be quasi-isometric.

R

Radius of metric space is the infimum of radii of metric balls which contain the space completely. Ray is a one side infinite geodesic which is minimizing on each interval. Ricci curvature Riemann Riemann curvature tensor Riemannian manifold Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.

S

Scalar curvature Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the shape operator of a hypersurface,

$\text{II}(v,w)=⟨S(v),w⟩$

It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space. Shape operator for a hypersurface M is a linear operator on tangent spaces, S_p: T_pM→T_pM. If n is a unit normal field to M and v is a tangent vector then

$S(v)=\pm {\mathrm{\nabla }}_{v}n$

(there is no standard agreement whether to use + or − in the definition). Short map is a distance non increasing map. Smooth manifold Sol manifold is a factor of a connected solvable Lie group by a lattice. Submetry A short map f between metric spaces is called a submetry if there exists R > 0 such that for any point x and radius r < R the image of metric r-ball is an r-ball, i.e.$$f(B_r(x))=B_r(f(x)).$$Sub-Riemannian manifold Systole. The k-systole of M, $sys{t}_k(M)$, is the minimal volume of k-cycle nonhomologous to zero.

T

Tangent bundle Totally convex A subset K of a Riemannian manifold M is called totally convex if for any two points in K any geodesic connecting them lies entirely in K, see also convex. Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.

U

Uniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic.

W

Word metric on a group is a metric of the Cayley graph constructed using a set of generators.