Glossary of Riemannian and metric geometry
Encyclopedia
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:
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.
Almost flat manifold
Arc-wise isometry the same as path isometry.
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
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.
Christoffel symbol
Collapsing manifold
Complete space
Completion
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.
Cotangent bundle
Covariant derivative
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
.
First fundamental form
for an embedding or immersion
is the pullback
of the metric tensor
.
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
.
Gromov-Hausdorff convergence
Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.
Horosphere a level set of Busemann function.
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.
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
on N.
A compact factor of N by subgroup of
acting freely on N is called infranil manifold.
Infranil manifolds are factors of nil manifolds by finite group (but the converse fails).
Isometry
is a map which preserves distances.
Intrinsic metric
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
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).
Lipschitz map
Logarithmic map is a right inverse of Exponential map.
Metric ball
Metric tensor
Minimal surface
is a submanifold with (vector of) mean curvature zero.
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.
Nil manifolds: the minimal set of manifolds which includes a point, and has the following property: any oriented
-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: 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
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.
Path isometry
Proper metric space is a metric space in which every closed ball
is compact
. Every proper metric space is complete
.
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 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
Riemann curvature tensor
Riemannian manifold
Riemannian submersion
is a map between Riemannian manifolds which is submersion
and submetry at the same time.
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.
Smooth manifold
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 for any point x and radius r we have that image of metric r-ball is an r-ball, i.e.
Sub-Riemannian manifold
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.
constructed using a set of generators.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
and metric geometry — it doesn't cover the terminology of differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
.
The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.
- ConnectionConnection (mathematics)In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
- CurvatureCurvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
- Metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
- Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
See also:
- Glossary of general topology
- Glossary of differential geometry and topologyGlossary of differential geometry and topologyThis is a glossary of terms specific to differential geometry and differential topology.The following two glossaries are closely related:*Glossary of general topology*Glossary of Riemannian and metric geometry.See also:*List of differential geometry topics...
- 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)Almost flat manifold
Arc-wise isometry the same as path isometry.
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 XBusemann function given a ray, γ :
C
Cartan–Hadamard theoremCartan–Hadamard theorem
The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point...
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
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
extended Einstein's General relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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 functionSuch a point is unique if all distances
are less than radius of convexity.Christoffel symbol
Collapsing manifold
Collapsing manifold
In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gn, such that as n goes to infinity the manifold is close to a k-dimensional space, where k In Riemannian geometry, a collapsing or collapsed manifold is an...
Complete space
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
Completion
Conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a 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
Conjugate points
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian.-Definition:...
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
Geodesic convexity
In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds...
. A function f on a Riemannian manifold is a convex if for any geodesic
the function is convexConvex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
. A function f is called
-convex if for any geodesic with natural parameter , the function is convexConvex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
.
Convex
Geodesic convexity
In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds...
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.
Cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
Covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
D
Diameter of a metric space is the supremum of distances between pairs of points.Developable surface
Developable surface
In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion . Conversely, it is a surface which can be made by transforming a plane...
is a surface isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
to the plane.
Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
.
F
Finsler metricFirst fundamental form
First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and...
for an embedding or immersion
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
is the pullback
Pullback
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
of the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
.
G
GeodesicGeodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
which locally minimizes distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
.
Geodesic flow is a flow
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
on a tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
TM of a manifold M, generated by a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
whose trajectories
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
are of the form
where is a geodesicGeodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
.
Gromov-Hausdorff convergence
Gromov-Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.-Gromov–Hausdorff distance:...
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 mapExponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
at p is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
. The injectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points. See also cut locus
Cut locus (Riemannian manifold)
In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances...
.
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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
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.
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
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
on N.A compact factor of N by subgroup of
acting freely on N is called infranil manifold.Infranil manifolds are factors of nil manifolds by finite group (but the converse fails).
Isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
is a map which preserves distances.
Intrinsic metric
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
J
Jacobi fieldJacobi field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space...
A Jacobi field is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
on a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
γ which can be obtained on the following way: Take a smooth one parameter family of geodesics
with , then the Jacobi field is described byJordan curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
L
Length metric the same as intrinsic metric.Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
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).
Lipschitz map
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
Logarithmic map is a right inverse of Exponential map.
M
Mean curvatureMean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
Metric ball
Metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
Minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
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
-net if for any point in X there is a point in S on the distance . This is distinct from topological netsNet (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
which generalise limits.
Nil manifolds: the minimal set of manifolds which includes a point, and has the following property: any oriented
-bundle over a nil manifold is a nil manifold. It also can be defined as a factor of a connected nilpotentNilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
by a lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
.
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
Parallel transportParallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
Polyhedral space
Polyhedral space
In mathematics, specifically in metric geometry, a polyhedral space is a certain metric space. A polyhedral space is a simplicial complex in which every simplex has a flat metric....
a simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
Principal curvature
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...
is the maximum and minimum normal curvatures at a point on a surface.
Principal direction is the direction of the principal curvatures.
Path isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
Proper metric space is a metric space in which every closed ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Every proper metric space is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
.
Q
Quasigeodesic has two meanings; here we give the most common. A map is called quasigeodesic if there are constants and such thatNote that a quasigeodesic is not necessarily a continuous curve.
Quasi-isometry
Quasi-isometry
In mathematics, a quasi-isometry is a means to compare the large-scale structure of metric spaces. The concept is especially important in Gromov's geometric group theory.-Definition:...
. A map
is called a quasi-isometry if there are constants and such thatand 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
Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
Riemannian submersion
Riemannian submersion
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces....
is a map between Riemannian manifolds which is submersion
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
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, 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
Short map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance .These maps are the morphisms in the category of metric spaces, Met ....
is a distance non increasing map.
Smooth manifold
Sol manifold is a factor of a connected solvable Lie group by a lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
.
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.
Sub-Riemannian manifold
Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces....
Systole. The k-systole of M,
, is the minimal volume of k-cycle nonhomologous to zero.T
Tangent bundleTangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
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.
W
Word metric on a group is a metric of the Cayley graphCayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
constructed using a set of generators.