# 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

Unless stated otherwise, letters *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 (the last one works only in dimension 2)

**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 : [0, ∞)→*X*, the Busemann function is defined by

## C

**Cartan–Hadamard theorem** is the statement that a connected, simply connected 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 (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* ∈ *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*.

**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*.

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

## F

**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.

**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* (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 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.

## 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

## 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(-*r*), exp(*r*).

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

## 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 (in ) of the tangent space .

**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** 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.

## Q

**Quasigeodesic** has two meanings; here we give the most common. A map (where is a subsegment) 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*(*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 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

(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 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.

**Systole**. The *k*-systole of *M*, , is the minimal volume of *k*-cycle nonhomologous to zero.

## T

**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.