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This is a glossary of some terms used in
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 that varies smoothly from point to poin ...
and
metric geometry In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
— it doesn't cover the terminology of
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
. The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below. * Connection *
Curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
*
Metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
*
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
See also: *
Glossary of general topology This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fund ...
*
Glossary of differential geometry and topology This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: *Glossary of general topology *Glossary of algebraic topology *Glossary of Riemannian and metric geometr ...
*
List of differential geometry topics This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Differential geometry of curves and surfaces Differential geometry of curves *List of curves topics *Frenet–Se ...
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. __NOTOC__


A

Alexandrov space In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces where t ...
a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2)
Almost flat manifold In mathematics, a smooth compact manifold ''M'' is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon on ''M'' such that \mbox(M,g_\varepsilon)\le 1 and g_\varepsilon is \varepsilon-flat, i.e. for the sect ...
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\le, f(x)f(y), _Y\le C, xy, _X
Busemann function In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named aft ...
given a '' ray'', γ : \gamma(t)-p, -t)


C

Cartan–Hadamard_theorem.html" ;"title="Conjugation
--> Cartan–Hadamard theorem">Conjugation --> 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. Élie Cartan, 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_ix, ^2 Such a point is unique if all distances , p_ip_j, are less than ''radius of convexity''.
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distan ...
Collapsing manifold In Riemannian geometry, a collapsing or collapsed manifold is an ''n''-dimensional manifold ''M'' that admits a sequence of Riemannian metrics ''g'i'', such that as ''i'' goes to infinity the manifold is close to a ''k''-dimensional space, w ...
Complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
Completion
Conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
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 In differential geometry, conjugate points or focal 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. Another viewpoi ...
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 In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
. A function ''f'' on a Riemannian manifold is a convex if for any geodesic \gamma the function f\circ\gamma is
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
. 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 or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
.
Convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
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 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. It may be described also as the dual bundle to the tangent bundle. This may ...
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 different ...
Cut locus The cut locus is a mathematical structure defined for a closed set S in a space X as the closure of the set of all points p\in X that have two or more distinct shortest paths in X from S to p. Definition in a special case Let X be a metric s ...


D

Diameter of a metric space is the supremum of distances between pairs of points.
Developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
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) In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of ...
,
Exponential map (Riemannian geometry) In Riemannian geometry, an exponential map is a map from a subset of a tangent space T''p'M'' of a Riemannian manifold (or pseudo-Riemannian manifold) ''M'' to ''M'' itself. The (pseudo) Riemannian metric determines a canonical affine connect ...


F

Finsler metric In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
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 . It permits the calculation of curvature and me ...
for an embedding or immersion is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...
of the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
.


G

Geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
which locally minimizes
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
.
Geodesic flow In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
is a flow on a
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''TM'' of a manifold ''M'', generated by a vector field whose
trajectories A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
are of the form (\gamma(t),\gamma'(t)) where \gamma is a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. Gromov-Hausdorff convergence Geodesic metric space is a metric space where any two points are the endpoints of a minimizing
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
.


H

Hadamard space In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete ...
is a complete simply connected space with nonpositive curvature.
Horosphere In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of ...
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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
. The injectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points. See also
cut locus The cut locus is a mathematical structure defined for a closed set S in a space X as the closure of the set of all points p\in X that have two or more distinct shortest paths in X from S to p. Definition in a special case Let X be a metric s ...
. For complete manifolds, if the injectivity radius at ''p'' is a finite number ''r'', then either there is a geodesic of length 2''r'' 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 In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
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 In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, th ...
.
Isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
is a map which preserves distances.
Intrinsic metric In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...


J

Jacobi 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 for ...
A Jacobi field is a vector field on a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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)=\left. \frac \_. Jordan curve


K

Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...


L

Length metric the same as ''intrinsic metric''.
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
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'').
Lipschitz map In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
Logarithmic map is a right inverse of Exponential map.


M

Mean 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. The ...
Metric ball
Metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
Minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
is a submanifold with (vector of) mean curvature zero.


N

Natural parametrization is the parametrization by length. Net. A subset ''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 \le\epsilon. This is distinct from topological nets which generalize limits.
Nilmanifold In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, th ...
: 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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
by a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
.
Normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...
: associated to an imbedding of a manifold ''M'' into an ambient Euclidean space ^N, the normal bundle is a vector bundle whose fiber at each point ''p'' is the orthogonal complement (in ^N) of the tangent space T_pM. Nonexpanding map same as ''short map''


P

Parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
Polyhedral space Polyhedral space is a certain metric space. A ( Euclidean) polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric. (Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every s ...
a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
with a metric such that each simplex with induced metric is isometric to a simplex in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
.
Principal curvature In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...
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 In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
. Equivalently, if every closed bounded subset is compact. Every proper metric space is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
.


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 \ge 1 and C \ge 0 such that for every x,y\in I :d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C. Note that a quasigeodesic is not necessarily a continuous curve.
Quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. T ...
. A map f:X\to Y is called a ''quasi-isometry'' if there are constants K \ge 1 and C \ge 0 such that :d(x,y)-C\le d(f(x),f(y))\le 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
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 common way used to express the curvature of Riemannian manifolds. ...
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
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. Formal definition Let (' ...
is a map between Riemannian manifolds which is submersion and ''submetry'' at the same time.


S

Second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
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(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 In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
for a hypersurface ''M'' is a linear operator on tangent spaces, ''S''''p'': ''T''''p''''M''→''T''''p''''M''. If ''n'' is a unit normal field to ''M'' and ''v'' is a tangent vector then :S(v)=\pm \nabla_n (there is no standard agreement whether to use + or − in the definition).
Short map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 1 ...
is a distance non increasing map.
Smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
Sol manifold In mathematics, a solvmanifold is a homogeneous space of a connected (topology), connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed (topology), closed subgroup. (Some authors also r ...
is a factor of a connected
solvable Lie group In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted : mathfrak,\mathfrak/math> that consist ...
by a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
. 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)) Sub-Riemannian manifold
Systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...
. The ''k''-systole of ''M'', syst_k(M), is the minimal volume of ''k''-cycle nonhomologous to zero.


T

Tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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 ''
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s'' 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
.


W

Word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that m ...
on a group is a metric of the
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...
constructed using a set of generators. {{DEFAULTSORT:Glossary Of Riemannian And Metric Geometry Differential geometry
Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
* *