# Glossary of Riemannian and metric geometry

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.

See also:

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

*X*,

*Y*,

*Z*below denote metric spaces,

*M*,

*N*denote Riemannian manifolds, |

*xy*| or 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

**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 is called bi-Lipschitz if there are positive constants

*c*and

*C*such that for any

*x*and

*y*in

*X*

**Busemann function**given a

*ray*, γ :

**R**

^{n}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 CAT space.

**Cartan**extended [Einstein's General relativity">Complete space">complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to

**R**

^{n}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 CAT 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*∈

*M*is called the center of mass of the points if it is a point of global minimum of the function

Such a point is unique if all distances are less than

*radius of convexity*.

**Christoffel symbol**

**Collapsing manifold**

**Complete space**

**Completion**

**Conformal map**is a map which preserves angles.

**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 are called

**conjugate**if there is a Jacobi field on which has a zero at

*p*and

*q*.

**Convex function.**A function

*f*on a Riemannian manifold is a convex if for any geodesic the function is convex. A function

*f*is called -convex if for any geodesic with natural parameter, the function is convex.

**Convex**A subset

*K*of a Riemannian manifold

*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*.

**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**of a map between metric spaces is the infimum of numbers

*L*such that the given map is

*L*-Lipschitz.

## E

**Exponential map**: Exponential map, Exponential map

## F

**Finsler metric**

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

## 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 where 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 largest radius 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 2

*r*which starts and ends at

*p*or there is a point

*q*conjugate to

*p*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 on

*N*. An orbit space of

*N*by a discrete subgroup of 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 with, then the Jacobi field is described by

**Jordan curve**

## K

**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 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, exp.

**Lipschitz map**

**Logarithmic map**is a right inverse of Exponential map.

## M

**Mean curvature**

**Metric ball**

**Metric tensor**

**Minimal surface**is a submanifold with mean curvature zero.

## N

**Natural parametrization**is the parametrization by length.

**Net**. A sub set

*S*of a metric space

*X*is called -net if for any point in

*X*there is a point in

*S*on the distance. 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 -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 imbedding of a manifold

*M*into an ambient Euclidean space, the normal bundle is a vector bundle whose fiber at each point

*p*is the orthogonal complement of the tangent space.

**Nonexpanding map**same as

*short map*

## P

**Parallel transport**

**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.

**Path isometry**

**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.

## Q

**Quasigeodesic**has two meanings; here we give the most common. A map is called a

*quasigeodesic*if there are constants and such that for every

Note that a quasigeodesic is not necessarily a continuous curve.

**Quasi-isometry.**A map is called a

*quasi-isometry*if there are constants and such that

and every point in

*Y*has distance at most

*C*from some point of

*f*.

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

**Riemann curvature tensor**

**Riemannian manifold**

**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 the

*shape operator*of a hypersurface,

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*

_{p}

*M*→

*T*

_{p}

*M*. If

*n*is a unit normal field to

*M*and

*v*is a tangent vector then

.

**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*we have that image of metric

*r*-ball is an

*r*-ball, i.e.

**Sub-Riemannian manifold**

**Systole**. The

*k*-systole of

*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.