Glossary of Riemannian and metric geometry

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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.

The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.

* Connection * Curvature * Metric space * Riemannian manifold See also: * 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 <math>|xy|_X</math> 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.

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== A ==

'''Affine connection'''

'''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''.

'''Asymptotic cone'''

'''Autoparallel''' the same as ''totally geodesic''.<ref>{{Cite book |last1=Kobayashi |first1=Shōshichi |author-link=Shoshichi Kobayashi |title=Foundations of differential geometry |last2=Nomizu |first2=Katsumi |author-link2=Katsumi Nomizu |date=1963 |publisher=Interscience Publishers, New York, NY |isbn=978-0-471-15732-8 |pages=53–62 |chapter=Chapter VII Submanifolds, 8. Autoparallel submanifolds and totally geodesic submanifolds |zbl=0175.48504}}</ref>

== B ==

'''Banach space'''

'''Barycenter''', see ''center of mass''.

'''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'' :<math>c|xy|_X\le|f(x)f(y)|_Y\le C|xy|_X.</math>

'''Boundary at infinity'''. In general, a construction that may be regarded as a space of directions at infinity. For geometric examples, see for instance hyperbolic boundary, Gromov boundary, visual boundary, Tits boundary, Thurston boundary. See also projective space and compactification.

'''Busemann function''' given a ''ray'', γ : <nowiki>[</nowiki>0, ∞)→''X'', the Busemann function is defined by<math display="block">B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)-p|-t).</math>

== C ==<!-- This section is linked from Conjugation -->

'''Cartan connection'''

'''Cartan-Hadamard space''' is a complete, simply-connected, non-positively curved Riemannian manifold.

'''Cartan–Hadamard theorem''' is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to '''R'''ⁿ 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)''' The mathematician after whom ''Cartan-Hadamard manifolds'', Cartan subalgebras, and ''Cartan connections'' are named (not to be confused with his son Henri Cartan).

<math display="inline">CAT(\kappa)</math> space

'''Center of mass'''. A point <math display="inline">q\in M</math> is called the center of mass<ref>{{Cite journal |last1=Mancinelli |first1=Claudio |last2=Puppo |first2=Enrico |date=2023-06-01 |title=Computing the Riemannian center of mass on meshes |journal=Computer Aided Geometric Design |volume=103 |article-number=102203 |doi=10.1016/j.cagd.2023.102203 |issn=0167-8396|doi-access=free }}</ref> of the points <math display="inline">p_1,p_2,\dots,p_k</math> if it is a point of global minimum of the function

:<math>f(x)=\sum_i |p_ix|^2.</math>

Such a point is unique if all distances <math>|p_ip_j|</math> are less than the ''convexity radius''.

'''Cheeger constant'''

'''Christoffel symbol'''

'''Coarse geometry'''

'''Collapsing manifold'''

'''Complete manifold''' According to the Riemannian Hopf-Rinow theorem, a Riemannian manifold is complete as a metric space, if and only if all geodesics can be infinitely extended.

'''Complete metric space'''

'''Completion'''

'''Complex hyperbolic space'''

'''Conformal map''' is a map which preserves angles.

'''Conformally flat''' a manifold ''M'' is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.

'''Conjugate points''' two points ''p'' and ''q'' on a geodesic <math>\gamma</math> are called '''conjugate''' if there is a Jacobi field on <math>\gamma</math> which has a zero at ''p'' and ''q''.

'''Connection'''

'''Convex function.''' A function ''f'' on a Riemannian manifold is a convex if for any geodesic <math>\gamma</math> the function <math>f\circ\gamma</math> is 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.

'''Convex''' A subset ''K'' of a Riemannian manifold ''M'' is called convex if for any two points in ''K'' there is a unique ''shortest path'' connecting them which lies entirely in ''K,'' see also ''totally convex''.

'''Convexity radius''' at a point <math display="inline">p</math> of a Riemannian manifold is the supremum of radii of balls centered at <math display="inline">p</math> that are ''(totally) convex''. The convexity radius of the manifold is the infimum of the convexity radii at its points; for a compact manifold this is a positive number.<ref>{{Citation |last1=Gallot |first1=Sylvestre |title=Riemannian metrics |date=2004 |work=Riemannian Geometry |editor-last=Gallot |editor-first=Sylvestre |url=https://link.springer.com/chapter/10.1007/978-3-642-18855-8_2 |access-date=2024-11-28 |at=Remark after Proof of Corollary 2.89, p.87 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-18855-8_2 |isbn=978-3-642-18855-8 |last2=Hulin |first2=Dominique |last3=Lafontaine |first3=Jacques |editor2-last=Hulin |editor2-first=Dominique |editor3-last=Lafontaine |editor3-first=Jacques|url-access=subscription }}</ref> Sometimes the additional requirement is made that the distance function to <math display="inline">p</math> in these balls is convex.<ref>{{Citation |last=Petersen |first=Peter |title=Sectional Curvature Comparison I |date=2016 |work=Riemannian Geometry |series=Graduate Texts in Mathematics |volume=171 |editor-last=Petersen |editor-first=Peter |url=https://link.springer.com/chapter/10.1007/978-3-319-26654-1_6 |access-date=2024-11-29 |at=Theorem 6.4.8, pp. 258-259 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-26654-1_6 |isbn=978-3-319-26654-1|url-access=subscription }}</ref>

'''Cotangent bundle'''

'''Covariant derivative'''

'''Cubical complex'''

'''Cut locus'''

== D ==

'''Diameter''' of a metric space is the supremum of distances between pairs of points.

'''Developable surface''' is a surface isometric to the plane.

'''Dilation''' same as ''Lipschitz constant''.

== E ==

'''Ehresmann connection'''

'''Einstein manifold'''

'''Euclidean geometry'''

'''Exponential map''' Exponential map (Lie theory), Exponential map (Riemannian geometry)

== F ==

'''Finsler metric''' A generalization of Riemannian manifolds where the scalar product on the tangent space is replaced by a norm.

'''First fundamental form''' for an embedding or immersion is the pullback of the metric tensor.

'''Flat manifold'''

== G ==

'''Geodesic''' is a curve which locally minimizes distance.

'''Geodesic equation''' is the differential equation whose local solutions are the geodesics.

'''Geodesic flow''' is a flow on a tangent bundle ''TM'' of a manifold ''M'', generated by a vector field whose trajectories are of the form <math>(\gamma(t),\gamma'(t))</math> where <math>\gamma</math> is a geodesic.

'''Gromov-Hausdorff convergence'''

'''Gromov-hyperbolic metric space'''

'''Geodesic metric space''' is a metric space where any two points are the endpoints of a minimizing geodesic.

== H ==

'''Hadamard space''' is a complete simply connected space with nonpositive curvature.

'''Hausdorff dimension'''

'''Hausdorff distance'''

'''Hausdorff measure'''

'''Hilbert space'''

'''Hölder map'''

'''Holonomy group''' is the subgroup of isometries of the tangent space obtained as ''parallel transport'' along closed curves.

'''Horosphere''' a level set of ''Busemann function''.

'''Hyperbolic geometry''' (see also ''Riemannian hyperbolic space'')

'''Hyperbolic link'''

== I ==

'''Injectivity radius''' The injectivity radius at a point ''p'' of a Riemannian manifold is the supremum of radii for which the exponential map at ''p'' is a diffeomorphism. The '''injectivity radius of a Riemannian manifold''' is the infimum of the injectivity radii at all points.<ref>{{Cite book |last=Lee |first=Jeffrey M. |title=Manifolds and differential geometry |date=2009 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=978-0-8218-4815-9 |pages=615 |language=en |chapter=13. Riemannian and Semi-Riemannian Geometry, Definition 13.141 |zbl=1190.58001}}</ref> 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 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''.<ref>{{Citation |last1=Gallot |first1=Sylvestre |title=Curvature |date=2004 |work=Riemannian Geometry |editor-last=Gallot |editor-first=Sylvestre |url=https://link.springer.com/chapter/10.1007/978-3-642-18855-8_3 |access-date=2024-11-28 |at=Scholium 3.78 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-18855-8_3 |isbn=978-3-642-18855-8 |last2=Hulin |first2=Dominique |last3=Lafontaine |first3=Jacques |editor2-last=Hulin |editor2-first=Dominique |editor3-last=Lafontaine |editor3-first=Jacques|url-access=subscription }}</ref> 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 <math>N \rtimes F</math> on ''N''. An orbit space of ''N'' by a discrete subgroup of <math display="inline">N \rtimes F</math> which acts freely on ''N'' is called an ''infranilmanifold''. An infranilmanifold is finitely covered by a nilmanifold.<ref>{{Cite book |last=Hirsch |first=Morris W. |title=Global Analysis |chapter=Expanding maps and transformation groups |series=Proceedings of Symposia in Pure Mathematics |date=1970 |volume=14 |issue=14 |pages=125–131 |doi=10.1090/pspum/014/0298701 |isbn=978-0-8218-1414-7 |zbl=0223.58009}}</ref>

'''Isometric embedding''' is an embedding preserving the Riemannian metric.

'''Isometry''' is a surjective map which preserves distances.

'''Isoperimetric function''' of a metric space <math display="inline">X</math> measures "how efficiently rectifiable loops are coarsely contractible with respect to their length". For the Cayley 2-complex of a finite presentation, they are equivalent to the Dehn function of the group presentation. They are invariant under quasi-isometries.<ref>{{Citation |last1=Bridson |first1=Martin R. |title=δ-Hyperbolic Spaces and Area |date=1999 |work=Metric Spaces of Non-Positive Curvature |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_21 |access-date=2024-12-23 |at=2. Area and isoperimetric inequalities, pp. 414 – 417 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_21 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André|url-access=subscription }}</ref>

'''Intrinsic metric'''

== J ==

'''Jacobi field''' A Jacobi field is a vector field on a geodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics <math>\gamma_\tau</math> with <math>\gamma_0=\gamma</math>, then the Jacobi field is described by :<math>J(t)=\left. \frac{\partial\gamma_\tau(t)}{\partial \tau} \right|_{\tau=0}.</math>

''' Jordan curve'''

== K ==

'''Kähler-Einstein metric'''

'''Kähler metric'''

'''Killing vector field'''

'''Koszul Connection'''

== L ==

'''Length metric''' the same as ''intrinsic metric''.

'''Length space'''

'''Levi-Civita connection''' is a natural way to differentiate vector fields on Riemannian manifolds.

'''Linear connection'''

'''Link'''

'''Lipschitz constant''' of a map is the infimum of numbers ''L'' such that the given map is ''L''-Lipschitz.

'''Lipschitz convergence''' the convergence of metric spaces defined by ''Lipschitz distance''.

'''Lipschitz distance''' between metric spaces is the infimum of numbers ''r'' such that there is a bijective ''bi-Lipschitz'' map between these spaces with constants exp(-''r''), exp(''r'').<ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergei |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |at=Chapter 7, §7.2, pp. 249-250 |zbl=0981.51016}}</ref>

'''Lipschitz map'''

'''Locally symmetric space'''

'''Logarithmic map''', or logarithm, is a right inverse of Exponential map.<ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergei |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |at=Chapter 9, §9.1, pp. 321-322 |zbl=0981.51016}}</ref><ref>{{Cite journal |last=Lang |first=Serge |date=1999 |title=Fundamentals of Differential Geometry |url=https://link.springer.com/book/10.1007/978-1-4612-0541-8 |journal=Graduate Texts in Mathematics |volume=191 |language=en |at=Chapter XII An example of seminegative curvature, p. 323 |doi=10.1007/978-1-4612-0541-8 |isbn=978-1-4612-6810-9 |issn=0072-5285|url-access=subscription }}</ref>

== M ==

'''Mean curvature'''

'''Metric ball'''

'''Metric tensor'''

'''Minkowski space'''

'''Minimal surface''' is a submanifold with (vector of) mean curvature zero.

'''Mostow's rigidity''' In dimension <math display="inline">\ge 3</math>, compact hyperbolic manifolds are classified by their fundamental group.

== N ==

'''Natural parametrization''' is the parametrization by length.<ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergei |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |at=Chapter 2, §2.5.1, Definition 2.5.7 |zbl=0981.51016}}</ref>

'''Net''' A subset ''S'' of a metric space ''X'' is called <math display="inline">\epsilon</math>-net if for any point in ''X'' there is a point in ''S'' on the distance <math display="inline">\le\epsilon</math>.<ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergei |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |at=Chapter 1, §1.6, Definition 1.6.1, p. 13 |zbl=0981.51016}}</ref> This is distinct from topological nets which generalize limits.

'''Nilmanifold''': An element of the minimal set of manifolds which includes a point, and has the following property: any oriented <math>S^1</math>-bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice.

'''Normal bundle''': associated to an embedding of a manifold ''M'' into an ambient Euclidean space <math display="inline">{\mathbb R}^N</math>, the normal bundle is a vector bundle whose fiber at each point ''p'' is the orthogonal complement (in <math display="inline">{\mathbb R}^N</math>) of the tangent space <math display="inline">T_pM</math>.

'''Nonexpanding map''' same as ''short map.''

== O == '''Orbifold'''

'''Orthonormal frame bundle''' is the bundle of bases of the tangent space that are orthonormal for the Riemannian metric.

== P ==

'''Parallel transport'''

'''Path isometry'''

'''Pre-Hilbert space'''

'''Polish space'''

'''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''.

'''Product metric'''

'''Product Riemannian manifold'''

'''Proper metric space''' is a metric space in which every closed ball is compact. Equivalently, if every closed bounded subset is compact. Every proper metric space is complete.<ref>{{Citation |last1=Bridson |first1=Martin R. |title=Basic Concepts |date=1999 |work=Metric Spaces of Non-Positive Curvature |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_1 |access-date=2024-11-29 |at=Chapter I.1, § Metric spaces, Definitions 1.1, p. 2 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_1 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André|url-access=subscription }}</ref>

'''Pseudo-Riemannian manifold'''

== Q ==

'''Quasi-convex subspace''' of a metric space <math display="inline">X</math> is a subset <math display="inline">Y\subseteq X</math> such that there exists <math display="inline">K\ge 0</math> such that for all <math display="inline">y, y'\in Y</math>, for all geodesic segment <math display="inline">[y, y']</math> and for all <math display="inline">z\in [y, y']</math>, <math display="inline">d(z, Y) \le K</math>.<ref>{{Citation |last1=Bridson |first1=Martin R. |title=Non-Positive Curvature and Group Theory |date=1999 |work=Metric Spaces of Non-Positive Curvature |editor-last=Bridson |editor-first=Martin R. |url=https://link.springer.com/chapter/10.1007/978-3-662-12494-9_22 |access-date=2024-12-23 |at=Definition 3.4, p. 460 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-12494-9_22 |isbn=978-3-662-12494-9 |last2=Haefliger |first2=André |editor2-last=Haefliger |editor2-first=André|url-access=subscription }}</ref>

'''Quasigeodesic''' has two meanings; here we give the most common. A map <math>f: I \to Y</math> (where <math> I\subseteq \mathbb R</math> is a subinterval) is called a ''quasigeodesic'' if there are constants <math>K \ge 1</math> and <math>C \ge 0</math> such that for every <math> x,y\in I</math> :<math>{1\over K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.</math> Note that a quasigeodesic is not necessarily a continuous curve.

'''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> 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.<ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergei |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |at=Chapter 10, §10.4, Exercise 10.4.5, p. 366 |zbl=0981.51016}}</ref>

'''Ray''' is a one side infinite geodesic which is minimizing on each interval.<ref>{{Cite journal |last=Petersen |first=Peter |date=2016 |title=Riemannian Geometry |url=https://link.springer.com/book/10.1007/978-3-319-26654-1 |journal=Graduate Texts in Mathematics |volume=171 |language=en |at=Chapter 7, §7.3.1 Rays and Lines, p. 298 |doi=10.1007/978-3-319-26654-1 |isbn=978-3-319-26652-7 |issn=0072-5285|url-access=subscription }}</ref>

'''Real tree'''

'''Rectifiable curve'''

'''Ricci curvature'''

'''Riemann''' The mathematician after whom ''Riemannian geometry'' is named.

'''Riemannian angle'''

'''Riemann curvature tensor''' is often defined as the (4, 0)-tensor of the tangent bundle of a Riemannian manifold <math display="inline">(M, g)</math> as<math display="block">R_p(X, Y, Z)W = {g_p({\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z, W})},</math>for <math display="inline">p\in M</math> and <math display="inline">X, Y, Z, W\in T_pM</math> (depending on conventions, <math display="inline">X</math> and <math display="inline">Y</math> are sometimes switched).

'''Riemannian hyperbolic space'''

'''Riemannian manifold'''

'''Riemannian submanifold''' A differentiable sub-manifold whose Riemannian metric is the restriction of the ambient Riemannian metric (not to be confused with ''sub-Riemannian manifold'').

'''Riemannian submersion''' is a map between Riemannian manifolds which is submersion and ''submetry'' at the same time.

== S ==

'''Scalar curvature'''

'''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, :<math>\text{II}(v,w)=\langle S(v),w\rangle.</math> It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.

'''Sectional curvature''' at a point <math display="inline">p</math> of a Riemannian manifold <math display="inline">M</math> along the 2-plane spanned by two linearly independent vectors <math display="inline">u, v\in T_pM</math> is the number<math display="block">\sigma_p({Vect}(u, v)) = \frac{R_p(u, v, v, u)}{g_p(u, u)g_p(v, v) - g_p(u, v)^2}</math>where <math display="inline">R_p</math> is the ''curvature tensor'' written as <math display="inline">R_p(X, Y, Z)W = {g_p({\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z, W})}</math>, and <math display="inline">{g_p}</math> is the Riemannian metric.

'''Shape operator''' for a hypersurface ''M'' is a linear operator on tangent spaces, ''S''_''p'':&nbsp;''T''_''p''''M''→''T''_''p''''M''. If ''n'' is a unit normal field to ''M'' and ''v'' is a tangent vector then :<math>S(v)=\pm \nabla_{v}n</math> (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.

'''Spherical geometry'''

'''Submetry''' A short map ''f'' between metric spaces is called a submetry<ref>{{Cite journal |last=Berestovskii |first=V. N. |date=1987-07-01 |title=Submetries of space-forms of negative curvature |url=https://link.springer.com/article/10.1007/BF00973842 |journal=Siberian Mathematical Journal |language=en |volume=28 |issue=4 |pages=552–562 |doi=10.1007/BF00973842 |bibcode=1987SibMJ..28..552B |issn=1573-9260|url-access=subscription }}</ref> if there exists ''R > 0'' such that for any point ''x'' and radius ''r < R'' the image of metric ''r''-ball is an ''r''-ball, i.e.<math display="block">f(B_r(x))=B_r(f(x)). </math>'''Sub-Riemannian manifold'''

'''Symmetric space''' Riemannian symmetric spaces are Riemannian manifolds in which the geodesic reflection at any point is an isometry. They turn out to be quotients of a real Lie group by a maximal compact subgroup whose Lie algebra is the fixed subalgebra of the involution obtained by differentiating the geodesic symmetry. This algebraic data is enough to provide a classification of the Riemannian symmetric spaces.

'''Systole''' The ''k''-systole of ''M'', <math display="inline">syst_k(M)</math>, is the minimal volume of ''k''-cycle nonhomologous to zero.

== T ==

'''Tangent bundle'''

'''Tangent cone'''

'''Thurston's geometries''' The eight 3-dimensional geometries predicted by Thurston's geometrization conjecture, proved by Perelman: <math display="inline">\mathbb{S}^3</math>, <math display="inline">\R\times\mathbb{S}^2</math>, <math display="inline">\mathbb{R}^3</math>, <math display="inline">\mathbb{R}\times \mathbb{H}^2</math>, <math display="inline">\mathbb{H}^3</math>, <math>\mathrm{Sol}</math>, <math>\mathrm{Nil}</math>, and <math display="inline">\widetilde{PSL}_2(\R)</math>.

'''Tits boundary'''

'''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''.<ref>{{Cite journal |last=Petersen |first=Peter |date=2016 |title=Riemannian Geometry |url=https://link.springer.com/book/10.1007/978-3-319-26654-1 |journal=Graduate Texts in Mathematics |volume=171 |language=en |at=Chapter 12, §12.4 The Soul Theorem, p. 463 |doi=10.1007/978-3-319-26654-1 |isbn=978-3-319-26652-7 |issn=0072-5285|url-access=subscription }}</ref>

'''Totally geodesic''' submanifold is a ''submanifold'' such that all ''geodesics'' in the submanifold are also geodesics of the surrounding manifold.<ref>{{Cite journal |last1=Gallot |first1=Sylvestre |last2=Hulin |first2=Dominique |last3=Lafontaine |first3=Jacques |date=2004 |title=Riemannian Geometry |url=https://link.springer.com/book/10.1007/978-3-642-18855-8 |journal=Universitext |language=en |at=Chapter 2, §2.C.1, Definition 2.80 bis, p.82 |doi=10.1007/978-3-642-18855-8 |isbn=978-3-540-20493-0 |issn=0172-5939|url-access=subscription }}</ref>

'''Tree-graded space'''

== U ==

'''Uniquely geodesic metric space''' is a metric space where any two points are the endpoints of a unique minimizing geodesic.

== V == '''Variation'''

'''Volume form'''

== W ==

'''Word metric''' on a group is a metric of the Cayley graph constructed using a set of generators.

== References == {{Reflist}}

{{Riemannian geometry}}

{{DEFAULTSORT:Glossary Of Riemannian And Metric Geometry}}

Category:Differential geometry Geometry * * Category:Wikipedia glossaries using unordered lists