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.

A

Affine connection
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.
Asymptotic cone
Autoparallel the same as totally geodesic.

B

Banach space
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
Boundary at infinity. In general, a construction that may be regarded as a space of directions at infinity. For geometric examples, see for instance hyperbolic boundary, Gromov boundary, visual boundary, Tits boundary, Thurston boundary. See also projective space and compactification.
Busemann function given a ray, γ : 0, ∞)→X, the Busemann function is defined by

C

Cartan connection
Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.
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 CAT space.
Cartan The mathematician after whom Cartan-Hadamard manifolds, Cartan subalgebras, and Cartan connections are named.
space
Center of mass. A point 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 the convexity radius.
Cheeger constant
Christoffel symbol
Coarse geometry
Collapsing manifold
Complete manifold According to the Riemannian Hopf-Rinow theorem, a Riemannian manifold is complete as a metric space, if and only if all geodesics can be infinitely extended.
Complete metric space
Completion
Complex hyperbolic space
Conformal map is a map which preserves angles.
Conformally flat a manifold 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.
Connection
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 unique shortest path connecting them which lies entirely in K, see also totally convex.
Convexity radius at a point of a Riemannian manifold is the supremum of radii of balls centered at that are convex. The convexity radius of the manifold is the infimum of the convexity radii at its points; for a compact manifold this is a positive number. Sometimes the additional requirement is made that the distance function to in these balls is convex.
Cotangent bundle
Covariant derivative
Cubical complex
'''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 same as Lipschitz constant.

E

Ehresmann connection
Einstein manifold
Euclidean geometry
Exponential map Exponential map, Exponential map

F

Finsler metric A generalization of Riemannian manifolds where the scalar product on the tangent space is replaced by a norm.
First fundamental form for an embedding or immersion is the pullback of the metric tensor.
'''Flat manifold'''

G

Geodesic is a curve which locally minimizes distance.
Geodesic equation is the differential equation whose local solutions are the geodesics.
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
Gromov-hyperbolic metric space
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.
Hausdorff dimension
Hausdorff distance
Hausdorff measure
Hilbert space
Hölder map
Holonomy group is the subgroup of isometries of the tangent space obtained as parallel transport along closed curves.
Horosphere a level set of Busemann function.
Hyperbolic geometry
'''Hyperbolic link'''

I

Injectivity radius The injectivity radius at a point p of a Riemannian manifold is the supremum of radii 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 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.
Isometric embedding is an embedding preserving the Riemannian metric.
Isometry is a surjective map which preserves distances.
Isoperimetric function of a metric space measures "how efficiently rectifiable loops are coarsely contractible with respect to their length". For the Cayley 2-complex of a finite presentation, they are equivalent to the Dehn function of the group presentation. They are invariant under quasi-isometries.
'''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

Kähler-Einstein metric
Kähler metric
Killing vector field
'''Koszul Connection'''

L

Length metric the same as intrinsic metric.
Length space
Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.
Linear connection
Link
Lipschitz constant of a map is the infimum of numbers L such that the given map is L-Lipschitz.
Lipschitz convergence the convergence of metric spaces defined by Lipschitz distance.
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
Locally symmetric space
Logarithmic map, or logarithm, is a right inverse of Exponential map.

M

Mean curvature
Metric ball
Metric tensor
Minkowski space
Minimal surface is a submanifold with mean curvature zero.
Mostow's rigidity In dimension, compact hyperbolic manifolds are classified by their fundamental group.

N

Natural parametrization is the parametrization by length.
Net A subset 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 embedding 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.''