# 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. These 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 $\displaystyle |xy|_X$ 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.

## 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.

## B

Baricenter, see center of mass.

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

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

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

$\displaystyle B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)p|-t)$

## C

Center of mass. A point q∈M is called the center of mass of the points $\displaystyle p_1,p_2,..,p_k$ if it is a point of global minimum of the function

$\displaystyle f(x)=\sum_i |p_ix|^2$

Such a point is unique if all distances $\displaystyle |p_ip_j|$ are less than radius of convexity.

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 $\displaystyle \gamma$ are called conjugate if there is a Jacobi field on $\displaystyle \gamma$ which has a zero at p and q.

Convex function. A function f on a Riemannian manifold is a convex if for any geodesic $\displaystyle \gamma$ the function $\displaystyle f\circ\gamma$ is convex. A function f is called $\displaystyle \lambda$ -convex if for any geodesic $\displaystyle \gamma$ with natural parameter $\displaystyle t$ , the function $\displaystyle f\circ\gamma(t)-\lambda t^2$ 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.

## 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.

## F

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

## G

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 $\displaystyle (\gamma(t),\gamma'(t))$ where $\displaystyle \gamma$ 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 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).

## 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 $\displaystyle \gamma_\tau$ with $\displaystyle \gamma_0=\gamma$ , then the Jacobi field

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

## L

Length metric the same as intrinsic metric.

Levi-Civita connection is a natural way to differentiate vector 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).

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 $\displaystyle \epsilon$ -net if for any point in X there is a point in S on the distance $\displaystyle \le\epsilon$ . 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 $\displaystyle S^1$ -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 direction

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 $\displaystyle f: \textbf{R} \to Y$ is called quasigeodesic if there are constants $\displaystyle K \ge 1$ and $\displaystyle C \ge 0$ such that

$\displaystyle {1\over K}|xy|-C\le |f(x)f(y)|\le K|xy|+C.$

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

Quasi-isometry. A map $\displaystyle f:X\to Y$ is called a quasi-isometry if there are constants $\displaystyle K \ge 1$ and $\displaystyle C \ge 0$ such that

$\displaystyle {1\over K}|xy|-C\le |f(x)f(y)|\le K|xy|+C.$

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,

$\displaystyle II(v,w)=\langle S(v),w\rangle$

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 $\displaystyle T_p(M)\to T_p(M)$ . If n is a unit normal field to M and v is a tangent vector then

$\displaystyle S(v)=\pm \nabla_{v}n$

(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.

$\displaystyle f(B_r(x))=B_r(f(x))$

Systole. k-systole of M, $\displaystyle syst_k(M)$ , 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.