Glossary of Riemannian and metric geometry

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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. These either contain specialised vocabulary or provide more detailed expositions of the definitions given below.

See also:

Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |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.

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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.

B

Baricenter, see center of mass.

bi-Lipschitz map. A map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)p|-t)}

C

Center of mass. A point q∈M is called the center of mass of the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1,p_2,..,p_k} if it is a point of global minimum of the function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\sum_i |p_ix|^2}

Such a point is unique if all distances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |p_ip_j|} are less than radius of convexity.

Complete space

Completion

Conformal map

Conformally flat a 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} are called conjugate if there is a Jacobi field on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ\gamma} is convex. A function f is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} -convex if for any geodesic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} with natural parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ\gamma(t)-\lambda t^2} is convex.

Convex A subset K of metric space M is called convex if for any two points in K there is a shortest path connecting them which lies entirely in K, see also totally convex.

Covariant derivative

D

Diameter of a metric space is the supremum of distances between pairs of points.

Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz.

E

Exponential map

F

Finsler metric

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

G

Geodesic

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\gamma(t),\gamma'(t))} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 at a point p of a Riemannian manifold is the largest radius for which the exponential map at p is a diffeomorphism. Injectivity radius of a Riemannian manifold is the infimum of Injectivity radii at all points.

For complete manifolds, if the injectivity radius at p is a finite, say 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 and on the distance r from p. For closed Riemannian manifold the injectivity radius is either half of minimal length of closed geodesic or minimal distance between conjugate points on a geodesic.

Infranil manifold 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 semidirect product NXF on N. A compact factor of N by subgroup of NXF acting freely on N is called infranil manifold. Infranil manifolds are factors of nil manifolds by finite group (but the converse fails).

Isometry

Intrinsic metric

J

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_\tau} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_0=\gamma} , then the Jacobi field

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}} .

Jordan curve

K

Killing vector field

L

Length metric the same as intrinsic metric.

Levi-Civita connection is a natural way to differentiate vector field on Riemannian manifolds.

Lipschitz convergence the convergence defined by Lipschitz metric.

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 is a right inverse of Exponential map

M

Metric ball

Minimal surface is a submanifold with (vector of) mean curvature zero.

N

Natural parametrization is the parametrization by length

Net. A sub set S of a metric space X is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} -net if for any point in X there is a point in S on the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \le\epsilon} . This is distinct from topological nets which generalise limits.

Nil manifolds: the minimal set of manifolds which includes a point, and has the following property: any oriented Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^1} -bundle over a nil manifold is a nil manifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice.

Normal bundle....

Nonexpanding map same as short map

P

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

Principal direction

Path isometry

Proper metric space is a metric space in which every closed ball is compact. Every proper metric space is complete.

Q

Quasigeodesic has two meanings here is the most common meaning. A map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \textbf{R} \to Y} is called quasigeodesic if there are constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \ge 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \ge 0} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1\over K}|xy|-C\le |f(x)f(y)|\le K|xy|+C.}

Note that a quasigeodesic is not a continuous curve in general.

Quasi-isometry. A map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to Y} is called a quasi-isometry if there are constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \ge 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \ge 0} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1\over K}|xy|-C\le |f(x)f(y)|\le K|xy|+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.

Radius of convexity at a point p of a Riemannian manifold is the largest radius of a ball which is a convex subset.

Ray is a one side infinite geodesic which is minimizing on each interval

Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.

S

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle II(v,w)=\langle S(v),w\rangle}

It can be also generalized to arbitrary codimension, then it is a quadratic form with values in the normal space.

Shape operator for a hypersurface M is a linear operator from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p(M)\to T_p(M)} . If n is a unit normal field to M and v is a tangent vector then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(v)=\pm \nabla_{v}n}

(there is no standard agreement whether to use + or - in the definition).

Short map is a distance non increasing map.

Sol manifold is a factor of a connected solvable Lie group by a lattice.

Submetry a short map f between metric spaces called submetry if for any point x and radius r we have that image of metric r-ball is an r-ball, i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(B_r(x))=B_r(f(x))}

Sub-Riemannian manifold

Systole. k-systole of M, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle syst_k(M)} , is the minimal volume of k-cycle nonhomologous to zero.

T

Totally convex. A subset K of metric space M is called totally convex if for any two points in K any shortest path 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.

W

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