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This is a glossary of some terms used inRiemannian geometry andmetric geometry — it doesn't cover the terminology ofdifferential 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, lettersX,Y,Z below denote metric spaces,M,N denote Riemannian manifolds, |xy| or${\displaystyle |xy{|}_X}$ denotes the distance between pointsx andy inX. Italicword denotes a self-reference to this glossary.

A caveat: many terms in Riemannian and metric geometry, such asconvex function,convex set and others, do not have exactly the same meaning as in general mathematical usage.

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 aspath isometry.

Asymptotic cone

Autoparallel the same astotally geodesic.^[1]

Banach space

Barycenter, seecenter of mass.

Bi-Lipschitz map. A map${\displaystyle f:X\to Y}$ is called bi-Lipschitz if there are positive constantsc andC such that for anyx andy inX

${\displaystyle c|xy{|}_X\le |f(x)f(y){|}_Y\le C|xy{|}_X.}$

Boundary at infinity. In general, a construction that may be regarded as a space of directions at infinity. For geometric examples, see for instancehyperbolic boundary,Gromov boundary,visual boundary,Tits boundary,Thurston boundary. See alsoprojective space andcompactification.

Busemann function given aray, γ : [0, ∞)→X, the Busemann function is defined by$${\displaystyle B_{\gamma }(p)=\underset{t\to \mathrm{\infty }}{lim}(|\gamma (t)-p|-t).}$$

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 toRⁿ 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 whomCartan-Hadamard manifolds,Cartan subalgebras, andCartan connections are named (not to be confused with his sonHenri Cartan).

$CAT(\kappa )$ space

Center of mass. A point$q\in M$ is called the center of mass^[2] of the points$p_1,p_2,\dots ,p_k$ if it is a point of global minimum of the function

${\displaystyle f(x)=\sum _i|p_{i}x{|}^2.}$

Such a point is unique if all distances${\displaystyle |p_i{p}_j|}$ are less than theconvexity radius.

Cheeger constant

Christoffel symbol

Coarse geometry

Collapsing manifold

Complete manifold According to the RiemannianHopf-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 manifoldM is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.

Conjugate points two pointsp andq on a geodesic${\displaystyle \gamma }$ are calledconjugate if there is a Jacobi field on${\displaystyle \gamma }$ which has a zero atp andq.

Connection

Convex function. A functionf on a Riemannian manifold is a convex if for any geodesic${\displaystyle \gamma }$ the function${\displaystyle f\circ \gamma }$ isconvex. A functionf is called${\displaystyle \lambda }$-convex if for any geodesic${\displaystyle \gamma }$ with natural parameter${\displaystyle t}$, the function${\displaystyle f\circ \gamma (t)-\lambda t^2}$ isconvex.

Convex A subsetK of a Riemannian manifoldM is called convex if for any two points inK there is a uniqueshortest path connecting them which lies entirely inK, see alsototally convex.

Convexity radius at a point$p$ of a Riemannian manifold is the supremum of radii of balls centered at$p$ 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.^[3] Sometimes the additional requirement is made that the distance function to$p$ in these balls is convex.^[4]

Cotangent bundle

Covariant derivative

Cubical complex

Cut locus

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

Developable surface is a surfaceisometric to the plane.

Dilation same asLipschitz constant.

Ehresmann connection

Einstein manifold

Euclidean geometry

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

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

First fundamental form for anembedding or immersion is thepullback of themetric tensor.

Flat manifold

Geodesic is acurve which locally minimizesdistance.

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

Geodesic flow is aflow on atangent bundleTM of a manifoldM, generated by avector field whosetrajectories are of the form${\displaystyle (\gamma (t),{\gamma }^{\prime }(t))}$ where${\displaystyle \gamma }$ is ageodesic.

Gromov-Hausdorff convergence

Gromov-hyperbolic metric space

Geodesic metric space is a metric space where any two points are the endpoints of a minimizinggeodesic.

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 asparallel transport along closed curves.

Horosphere a level set ofBusemann function.

Hyperbolic geometry (see alsoRiemannian hyperbolic space)

Hyperbolic link

Injectivity radius The injectivity radius at a pointp of a Riemannian manifold is the supremum of radii for which theexponential map atp is adiffeomorphism. Theinjectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points.^[5] See alsocut locus.

For complete manifolds, if the injectivity radius atp is a finite numberr, then either there is a geodesic of length 2r which starts and ends atp or there is a pointq conjugate top (seeconjugate point above) and on the distancer fromp.^[6] For aclosed 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 groupN acting on itself by left multiplication and a finite group of automorphismsF ofN one can define an action of thesemidirect product${\displaystyle N⋊F}$ onN. An orbit space ofN by a discrete subgroup of$N⋊F$ which acts freely onN is called aninfranilmanifold. An infranilmanifold is finitely covered by anilmanifold.^[7]

Isometric embedding is an embedding preserving the Riemannian metric.

Isometry is a surjective map which preserves distances.

Isoperimetric function of a metric space$X$ 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 theDehn function of the group presentation. They are invariant under quasi-isometries.^[8]

Intrinsic metric

Jacobi field A Jacobi field is avector field on ageodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics${\displaystyle {\gamma }_{\tau }}$ with${\displaystyle {\gamma }_0=\gamma }$, then the Jacobi field is described by

${\displaystyle J(t)={\frac{\mathrm{\partial }{\gamma }_{\tau }(t)}{\mathrm{\partial }\tau }|}_{\tau =0}.}$

Jordan curve

Kähler-Einstein metric

Kähler metric

Killing vector field

Koszul Connection

Length metric the same asintrinsic 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 numbersL such that the given map isL-Lipschitz.

Lipschitz convergence the convergence of metric spaces defined byLipschitz distance.

Lipschitz distance between metric spaces is the infimum of numbersr such that there is a bijectivebi-Lipschitz map between these spaces with constants exp(-r), exp(r).^[9]

Lipschitz map

Locally symmetric space

Logarithmic map, or logarithm, is a right inverse of Exponential map.^[10][11]

Mean curvature

Metric ball

Metric tensor

Minkowski space

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

Mostow's rigidity In dimension$\ge 3$, compact hyperbolic manifolds are classified by their fundamental group.

Natural parametrization is the parametrization bylength.^[12]

Net A subsetS of a metric spaceX is called$ϵ$-net if for any point inX there is a point inS on the distance$\le ϵ$.^[13] This is distinct fromtopological nets which generalize limits.

Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented${\displaystyle S^1}$-bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connectednilpotentLie group by alattice.

Normal bundle: associated to an embedding of a manifoldM into an ambient Euclidean space${\mathbb{R}}^N$, the normal bundle is a vector bundle whose fiber at each pointp is the orthogonal complement (in${\mathbb{R}}^N$) of the tangent space$T_{p}M$.

Nonexpanding map same asshort map.

Orbifold

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

Parallel transport

Path isometry

Pre-Hilbert space

Polish space

Polyhedral space asimplicial complex with a metric such that each simplex with induced metric is isometric to a simplex inEuclidean space.

Principal curvature is the maximum and minimum normal curvatures at a point on a surface.

Principal direction is the direction of theprincipal curvatures.

Product metric

Product Riemannian manifold

Proper metric space is a metric space in which everyclosed ball iscompact. Equivalently, if every closed bounded subset is compact. Every proper metric space iscomplete.^[14]

Pseudo-Riemannian manifold

Quasi-convex subspace of a metric space$X$ is a subset$Y\subseteq X$ such that there exists$K\ge 0$ such that for all$y,y^{\prime }\in Y$, for all geodesic segment$[y,y^{\prime }]$ and for all$z\in [y,y^{\prime }]$,$d(z,Y)\le K$.^[15]

Quasigeodesic has two meanings; here we give the most common. A map${\displaystyle f:I\to Y}$ (where${\displaystyle I\subseteq \mathbb{R}}$ is a subinterval) is called aquasigeodesic if there are constants${\displaystyle K\ge 1}$ and${\displaystyle C\ge 0}$ such that for every${\displaystyle x,y\in I}$

${\displaystyle \frac{1}{K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.}$

Note that a quasigeodesic is not necessarily a continuous curve.

Quasi-isometry. A map${\displaystyle f:X\to Y}$ is called aquasi-isometry if there are constants${\displaystyle K\ge 1}$ and${\displaystyle C\ge 0}$ such that

${\displaystyle \frac{1}{K}d(x,y)-C\le d(f(x),f(y))\le Kd(x,y)+C.}$

and every point inY has distance at mostC from some point off(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 bequasi-isometric.

Radius of metric space is the infimum of radii of metric balls which contain the space completely.^[16]

Ray is a one side infinite geodesic which is minimizing on each interval.^[17]

Real tree

Rectifiable curve

Ricci curvature

Riemann The mathematician after whomRiemannian geometry is named.

Riemannian angle

Riemann curvature tensor is often defined as the (4, 0)-tensor of the tangent bundle of a Riemannian manifold$(M,g)$ as$${\displaystyle R_p(X,Y,Z)W=g_p({\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z,W),}$$for$p\in M$ and$X,Y,Z,W\in T_{p}M$ (depending on conventions,$X$ and$Y$ 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 withsub-Riemannian manifold).

Riemannian submersion is a map between Riemannian manifolds which issubmersion andsubmetry at the same time.

Scalar curvature

Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe theshape operator of a hypersurface,

${\displaystyle \text{II}(v,w)=⟨S(v),w⟩.}$

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$p$ of a Riemannian manifold$M$ along the 2-plane spanned by two linearly independent vectors$u,v\in T_{p}M$ is the number$${\displaystyle {\sigma }_p(Vect(u,v))=\frac{R_p(u,v,v,u)}{g_p(u,u)g_p(v,v)-g_p(u,v{)}^2}}$$where$R_p$ is thecurvature tensor written as$R_p(X,Y,Z)W=g_p({\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z,W)$, and$g_p$ is the Riemannian metric.

Shape operator for a hypersurfaceM is a linear operator on tangent spaces,S_p: T_pM→T_pM. Ifn is a unit normal field toM andv is a tangent vector then

${\displaystyle S(v)=\pm {\mathrm{\nabla }}_{v}n}$

(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 connectedsolvable Lie group by alattice.

Spherical geometry

Submetry A short mapf between metric spaces is called a submetry^[18] if there existsR > 0 such that for any pointx and radiusr < R the image of metricr-ball is anr-ball, i.e.$${\displaystyle f(B_r(x))=B_r(f(x)).}$$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 Thek-systole ofM,$sys{t}_k(M)$, is the minimal volume ofk-cycle nonhomologous to zero.

Tangent bundle

Tangent cone

Thurston's geometries The eight 3-dimensional geometries predicted byThurston's geometrization conjecture, proved by Perelman:${\mathbb{S}}^3$,$\mathbb{R}\times {\mathbb{S}}^2$,${\mathbb{R}}^3$,$\mathbb{R}\times {\mathbb{H}}^2$,${\mathbb{H}}^3$,${\displaystyle \mathrm{S}\mathrm{o}\mathrm{l}}$,${\displaystyle \mathrm{N}\mathrm{i}\mathrm{l}}$, and${\widetilde{PSL}}_2(\mathbb{R})$.

Tits boundary

Totally convex A subsetK of a Riemannian manifoldM is called totally convex if for any two points inK any geodesic connecting them lies entirely inK, see alsoconvex.^[19]

Totally geodesic submanifold is asubmanifold such that allgeodesics in the submanifold are also geodesics of the surrounding manifold.^[20]

Tree-graded space

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

Variation

Volume form

Word metric on a group is a metric of theCayley graph constructed using a set of generators.

• Differential geometry
• Glossaries of mathematics
• Metric geometry
• Riemannian geometry

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