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| {{Redirect|Radius of convexity|the anatomical feature of the [[Radius (bone)|radius bone]]|Convexity of radius}}
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| {{Unreferenced|date=December 2009}}
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| This is a glossary of some terms used in [[Riemannian geometry]] and [[metric geometry]] — it doesn't cover the terminology of [[differential topology]].
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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.
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| * [[Connection (mathematics)|Connection]]
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| * [[Curvature]]
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| * [[Metric space]]
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| * [[Riemannian manifold]]
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| See also:
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| * [[Glossary of general topology]]
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| * [[Glossary of differential geometry and topology]]
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| * [[List of differential geometry topics]]
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| Unless stated otherwise, letters ''X'', ''Y'', ''Z'' below denote metric spaces, ''M'', ''N'' denote Riemannian manifolds, |''xy''| or <math>|xy|_X</math> denotes the distance between points ''x'' and ''y'' in ''X''. Italic ''word'' denotes a self-reference to this glossary.
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| ''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.
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| {{compactTOC8|side=yes|top=yes|num=yes}}
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| __NOTOC__
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| == A ==
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| '''Alexandrov space''' a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2)
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| '''[[Almost flat manifold]]'''
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| '''Arc-wise isometry''' the same as ''path isometry''.
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| == B ==
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| '''Barycenter''', see ''center of mass''.
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| '''bi-Lipschitz map.''' A map <math>f:X\to Y</math> is called bi-Lipschitz if there are positive constants ''c'' and ''C'' such that for any ''x'' and ''y'' in ''X''
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| :<math>c|xy|_X\le|f(x)f(y)|_Y\le C|xy|_X</math>
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| '''Busemann function''' given a ''ray'', γ : <nowiki>[</nowiki>0, ∞)→''X'', the Busemann function is defined by
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| :<math>B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)-p|-t)</math>
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| == C ==<!-- This section is linked from [[Conjugation]] -->
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| '''[[Cartan–Hadamard theorem]]''' is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to '''R'''<sup>n</sup> 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]]. | |
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| '''[[Élie Cartan|Cartan]]''' extended Einstein's [[General relativity]] to [[Einstein-Cartan theory]], using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides [[Torsion (differential geometry)|affine torsion]], which allows for non-symmetric curvature tensors and the incorporation of [[spin-orbit coupling]].
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| '''Center of mass'''. A point ''q'' ∈ ''M'' is called the center of mass of the points <math>p_1,p_2,\dots,p_k</math> if it is a point of global minimum of the function
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| :<math>f(x)=\sum_i |p_ix|^2</math>
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| Such a point is unique if all distances <math>|p_ip_j|</math> are less than ''radius of convexity''.
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| '''[[Christoffel symbol]]'''
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| '''[[Collapsing manifold]]'''
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| '''[[Complete space]]'''
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| '''[[Complete space#Completion|Completion]]'''
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| '''[[Conformal map]]''' is a map which preserves angles.
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| '''Conformally flat''' a ''M'' is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.
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| '''[[Conjugate points]]''' two points ''p'' and ''q'' on a geodesic <math>\gamma</math> are called '''conjugate''' if there is a Jacobi field on <math>\gamma</math> which has a zero at ''p'' and ''q''.
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| '''[[Geodesic convexity|Convex function]].''' A function ''f'' on a Riemannian manifold is a convex if for any geodesic <math>\gamma</math> the function <math>f\circ\gamma</math> is [[Convex function|convex]]. A function ''f'' is called <math>\lambda</math>-convex if for any geodesic <math>\gamma</math> with natural parameter <math>t</math>, the function <math>f\circ\gamma(t)-\lambda t^2</math> is [[Convex function|convex]].
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| '''[[Geodesic convexity|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''.
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| '''[[Cotangent bundle]]'''
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| '''[[Covariant derivative]]'''
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| '''[[Cut locus]]'''
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| == D ==
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| '''Diameter''' of a metric space is the supremum of distances between pairs of points.
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| '''[[Developable surface]]''' is a surface [[isometry|isometric]] to the plane.
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| '''Dilation''' of a map between metric spaces is the infimum of numbers ''L'' such that the given map is ''L''-[[Lipschitz continuity|Lipschitz]].
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| == E ==
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| '''[[Exponential map]]'''
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| == F ==
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| '''[[Finsler metric]]'''
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| '''[[First fundamental form]]''' for an [[Embedding|embedding or immersion]] is the [[pullback]] of the [[metric tensor]].
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| == G ==
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| '''[[Geodesic]]''' is a [[curve]] which locally minimizes [[distance]].
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| '''[[Geodesic flow]]''' is a [[Flow (mathematics)|flow]] on a [[tangent bundle]] ''TM'' of a manifold ''M'', generated by a [[vector field]] whose [[trajectory|trajectories]] are of the form <math>(\gamma(t),\gamma'(t))</math> where <math>\gamma</math> is a [[geodesic]].
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| [[Gromov-Hausdorff convergence]]
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| '''Geodesic metric space''' is a metric space where any two points are the endpoints of a minimizing [[geodesic#Metric geometry|geodesic]].
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| == H ==
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| '''Hadamard space''' is a complete simply connected space with nonpositive curvature.
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| '''[[Horosphere]]''' a level set of ''Busemann function''.
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| == I ==
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| '''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 (Riemannian manifold)|cut locus]].
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| 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
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| at ''p'' or there is a point ''q'' conjugate to ''p'' (see '''conjugate point''' above) and on the distance ''r'' from ''p''. For a [[manifold|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.
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| '''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]] <math>N \rtimes F</math> on ''N''.
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| An orbit space of ''N'' by a discrete subgroup of <math>N \rtimes F</math> which acts freely on ''N'' is called an ''infranilmanifold''.
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| An infranilmanifold is finitely covered by a nilmanifold.
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| '''[[Isometry]]''' is a map which preserves distances.
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| '''[[Intrinsic metric]]'''
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| == J ==
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| '''[[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 <math>\gamma_\tau</math> with <math>\gamma_0=\gamma</math>, then the Jacobi field is described by
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| :<math>J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}.\,</math>
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| '''[[Curve|Jordan curve]]'''
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| == K == | |
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| '''[[Killing vector field]]'''
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| == L ==
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| '''Length metric''' the same as ''intrinsic metric''.
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| '''[[Levi-Civita connection]]''' is a natural way to differentiate vector fields on Riemannian manifolds.
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| '''Lipschitz convergence''' the convergence defined by Lipschitz metric.
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| '''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'').
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| '''[[Lipschitz continuity|Lipschitz map]]'''
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| '''Logarithmic map''' is a right inverse of Exponential map.
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| == M ==
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| '''[[Mean curvature]]'''
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| '''Metric ball'''
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| '''[[Metric tensor]]'''
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| '''[[Minimal surface]]''' is a submanifold with (vector of) mean curvature zero.
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| == N ==
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| '''Natural parametrization''' is the parametrization by length.
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| '''Net'''. A sub set ''S'' of a metric space ''X'' is called <math> \epsilon</math>-net if for any point in ''X'' there is a point in ''S'' on the distance <math>\le\epsilon</math>. This is distinct from [[Net (mathematics)|topological nets]] which generalise limits.
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| '''[[Nilmanifold]]''': An element of the minimal set of manifolds which includes a point, and has the following property: any oriented <math>S^1</math>-bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected [[Nilpotent group|nilpotent]] [[Lie group]] by a [[lattice (discrete subgroup)|lattice]].
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| '''Normal bundle''': associated to an imbedding of a manifold ''M'' into an ambient Euclidean space <math>{\mathbb R}^N</math>, the normal bundle is a vector bundle whose fiber at each point ''p'' is the orthogonal complement (in <math>{\mathbb R}^N</math>) of the tangent space <math>T_pM</math>.
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| '''Nonexpanding map''' same as ''short map''
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| == P ==
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| '''[[Parallel transport]]'''
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| '''[[Polyhedral space]]''' a [[simplicial complex]] with a metric such that each simplex with induced metric is isometric to a simplex in [[Euclidean space]].
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| '''[[Principal curvature]]''' is the maximum and minimum normal curvatures at a point on a surface.
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| '''Principal direction''' is the direction of the principal curvatures.
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| '''[[Isometry|Path isometry]]'''
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| '''Proper metric space''' is a metric space in which every [[Ball (mathematics)|closed ball]] is [[compact space|compact]]. Every proper metric space is [[Complete space|complete]].
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| == Q ==
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| '''Quasigeodesic''' has two meanings; here we give the most common. A map <math>f: \textbf{R} \to Y</math> is called quasigeodesic if there are constants <math>K > 0</math> and <math>C \ge 0</math> such that
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| :<math>{1\over K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.</math>
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| Note that a quasigeodesic is not necessarily a continuous curve.
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| '''[[Quasi-isometry]].''' A map <math>f:X\to Y</math> is called a quasi-isometry if there are constants <math>K \ge 1</math> and <math>C \ge 0</math> such that | |
| :<math>{1\over K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.</math>
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| and every point in ''Y'' has distance at most ''C'' from some point of ''f''(''X'').
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| 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'''.
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| == R ==
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| '''Radius''' of metric space is the infimum of radii of metric balls which contain the space completely.
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| '''Radius of convexity''' at a point ''p'' of a Riemannian manifold is the largest radius of a ball which is a ''convex'' subset.
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| '''Ray''' is a one side infinite geodesic which is minimizing on each interval
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| '''[[Riemann curvature tensor]]'''
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| '''[[Riemannian manifold]]'''
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| '''[[Riemannian submersion]]''' is a map between Riemannian manifolds which is [[submersion (mathematics)|submersion]] and ''submetry'' at the same time.
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| == S ==
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| '''[[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,
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| :<math>\text{II}(v,w)=\langle S(v),w\rangle</math>
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| It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.
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| '''Shape operator''' for a hypersurface ''M'' is a linear operator on tangent spaces, ''S''<sub>''p''</sub>: ''T''<sub>''p''</sub>''M''→''T''<sub>''p''</sub>''M''. If ''n'' is a unit normal field to ''M'' and ''v'' is a tangent vector then
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| :<math>S(v)=\pm \nabla_{v}n</math>
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| (there is no standard agreement whether to use + or − in the definition).
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| '''[[Short map]]''' is a distance non increasing map.
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| '''[[Smooth manifold]]'''
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| '''[[Sol manifold]]''' is a factor of a connected [[solvable Lie group]] by a [[lattice (discrete subgroup)|lattice]].
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| '''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.
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| :<math>f(B_r(x))=B_r(f(x)) \,\!</math>
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| '''[[Sub-Riemannian manifold]]'''
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| '''[[systolic geometry|Systole]]'''. The ''k''-systole of ''M'', <math>syst_k(M)</math>, is the minimal volume of ''k''-cycle nonhomologous to zero.
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| == T ==
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| '''[[Tangent bundle]]'''
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| '''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''.
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| '''Totally geodesic''' submanifold is a ''submanifold'' such that all ''[[geodesic]]s'' in the submanifold are also geodesics of the surrounding manifold.
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| == U ==
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| '''Uniquely geodesic metric space''' is a metric space where any two points are the endpoints of a unique minimizing [[geodesic#Metric geometry|geodesic]].
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| == W ==
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| '''[[Word metric]]''' on a group is a metric of the [[Cayley graph]] constructed using a set of generators.
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| {{DEFAULTSORT:Glossary Of Riemannian And Metric Geometry}}
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| [[Category:Glossaries of mathematics|Geometry]]
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| [[Category:Metric geometry|*]]
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| [[Category:Riemannian geometry|*]]
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