# 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.

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)

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

$c|xy|_{X}\leq |f(x)f(y)|_{Y}\leq C|xy|_{X}$ Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by

$B_{\gamma }(p)=\lim _{t\to \infty }(|\gamma (t)-p|-t)$ ## 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 ∈ 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 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 $\gamma$ 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 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 $(\gamma (t),\gamma '(t))$ where $\gamma$ 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 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 $N\rtimes F$ on N. An orbit space of N by a discrete subgroup of $N\rtimes F$ which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold.

Isometry is a map which preserves distances.

## 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

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

## 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 $\epsilon$ -net if for any point in X there is a point in S on the distance $\leq \epsilon$ . 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 imbedding 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

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 $f:I\to Y$ (where $I\subseteq \mathbb {R}$ is a subsegment) is called a quasigeodesic if there are constants $K\geq 1$ and $C\geq 0$ such that for every $x,y\in I$ ${1 \over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.$ 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 $K\geq 1$ and $C\geq 0$ such that

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

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,

${\text{II}}(v,w)=\langle S(v),w\rangle$ 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, SpTpMTpM. If n is a unit normal field to M and v is a tangent vector then

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

$f(B_{r}(x))=B_{r}(f(x))$ Systole. The k-systole of M, $syst_{k}(M)$ , 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.