Glossary of Riemannian and metric geometry
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The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.
- 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.
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.
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
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 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.
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.
Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.
Hadamard space is a complete simply connected space with nonpositive curvature.
Horosphere a level set of Busemann function.
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 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.
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.
Minimal surface is a submanifold with (vector of) mean curvature zero.
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 generalise 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
Principal curvature is the maximum and minimum normal curvatures at a point on a surface.
Principal direction is the direction of the principal curvatures.
Quasigeodesic has two meanings; here we give the most common. A map is called quasigeodesic if there are constants and such that
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.
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
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, Sp: TpM→TpM. 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.
Submetry a short map f between metric spaces is called a submetry if for any point x and radius 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.
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.