- Glossary of Riemannian and metric geometry
This is a glossary of some terms used in

Riemannian geometry andmetric geometry — it doesn't cover the terminology ofdifferential topology . The following articles may also be useful. These 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 $|xy|\_X$ 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.

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**A****Alexandrov space**a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2)Almost flat manifold **Arc-wise isometry**the same as "path isometry".**B****Baricenter**, see "center of mass".**bi-Lipschitz map.**A map $f:X\; o\; Y$ is called bi-Lipschitz if there are positive constants "c" and "C" such that for any "x" and "y" in "X":$c|xy|\_Xle|f(x)f(y)|\_Yle\; C|xy|\_X$**Busemann function**given a "ray", γ :[ 0, ∞)→"X", the Busemann function is defined by:$B\_gamma(p)=lim\_\{t\; oinfty\}(|gamma(t)p|-t)$**C**is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic toCartan–Hadamard theorem **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.**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 Collapsing manifold Complete space **Completion**is a map which preserves angles.Conformal map **Conformally flat**a "M" is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.two points "p" and "q" on a geodesic $gamma$ are calledConjugate points **conjugate**if there is a Jacobi field on $gamma$ which has a zero at "p" and "q".**Convex function.**A function "f" on a Riemannian manifold is a convex if for any geodesic $gamma$ the function $fcircgamma$ is convex. A function "f" is called $lambda$-convex if for any geodesic $gamma$ with natural parameter $t$, the function $fcircgamma(t)-lambda\; t^2$ is convex.**Convex**A subset "K" of a Riemannian manifold "M" is called convex if for any two points in "K" there is a "shortest path" connecting them which lies entirely in "K", see also "totally convex".Cotangent bundle Covariant derivative **D****Diameter**of a metric space is the supremum of distances between pairs of points.is a surface isometric to the plane.Developable surface **Dilation**of a map between metric spaces is the infimum of numbers "L" such that the given map is "L"-Lipschitz.**E**Exponential map **F**Finsler metric for an embedding or immersion is theFirst fundamental form pullback of themetric tensor .**G**is aGeodesic curve which locally minimizesdistance .is aGeodesic flow flow on atangent bundle "TM" of a manifold "M", generated by avector field whose trajectories are of the form $(gamma(t),gamma\text{'}(t))$ where $gamma$ is ageodesic .Gromov-Hausdorff convergence **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.a level set of "Busemann function".Horosphere **I****Injectivity radius**The injectivity radius at a point "p" of a Riemannian manifold is the largest radius for which theexponential map at "p" is adiffeomorphism . The**injectivity radius of a Riemannian manifold**is the infimum of the injectivity radii at all points.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.**Infranil manifold**Given a simply connected nilpotent Lie group "N" acting on itself by left multiplication and a finite group of automorphisms "F" of "N" one can define an action ofsemidirect product $N\; times\; F$ on "N". A compact factor of "N" by subgroup of $N\; times\; F$ acting freely on "N" is called "infranil manifold".Infranil manifolds are factors of nil manifolds by finite group (but the converse fails).is a map which preserves distances.Isometry Intrinsic metric **J**A Jacobi field is aJacobi field vector field on ageodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics $gamma\_\; au$ with $gamma\_0=gamma$, then the Jacobi field is described by:$J(t)=partialgamma\_\; au(t)/partial\; au|\_\{\; au=0\}$.**K**Killing vector field **L****Length metric**the same as "intrinsic metric".is a natural way to differentiate vector field on Riemannian manifolds.Levi-Civita connection **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").**Logarithmic map**is a right inverse of Exponential map.**M**Mean curvature **Metric ball**Metric tensor is a submanifold with (vector of) mean curvature zero.Minimal surface **N****Natural parametrization**is the parametrization by length.**Net**. A sub set "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 $leepsilon$. This is distinct from topological nets which generalise limits.: the minimal set of manifolds which includes a point, and has the following property: any oriented $S^1$-bundle over a nil manifold is a nil manifold. It also can be defined as a factor of a connected nilpotentNil manifold sLie group by a lattice.**Normal bundle**: associated to an imbedding of a manifold "M" into an ambient Euclidean space $\{mathbb\; R\}^N$, the normal bundle is a vector bundle whose fiber at each point "p" is the orthogonal complement (in $\{mathbb\; R\}^N$) of the tangent space $T\_pM$.**Nonexpanding map**same as "short map"**P**Parallel transport aPolyhedral space simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex inEuclidean space .is the maximum and minimum normal curvatures at a point on a surface.Principal curvature **Principal direction**is the direction of the principal curvatures.**Proper metric space**is a metric space in which every closed ball is compact. Every proper metric space is complete.**Q****Quasigeodesic**has two meanings; here we give the most common. A map $f:\; extbf\{R\}\; o\; Y$ is called quasigeodesic if there are constants $K\; >\; 0$ and $C\; ge\; 0$ such that :$\{1over\; K\}d(x,y)-Cle\; d(f(x),f(y))le\; Kd(x,y)+C.$ Note that a quasigeodesic is not necessarily a continuous curve.**Quasi-isometry.**A map $f:X\; o\; Y$ is called a quasi-isometry if there are constants $K\; ge\; 1$ and $C\; ge\; 0$ such that :$\{1over\; K\}d(x,y)-Cle\; d(f(x),f(y))le\; Kd(x,y)+C.$ 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 intervalRiemann curvature tensor Riemannian manifold is a map between Riemannian manifolds which is submersion and "submetry" at the same time.Riemannian submersion **S**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, :$II(v,w)=langle\; S(v),w\; angle$It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.Second fundamental form **Shape operator**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 :$S(v)=pm\; abla\_\{v\}n$ (there is no standard agreement whether to use + or − in the definition).is a distance non increasing map.Short map Smooth manifold is a factor of a connectedSol manifold solvable Lie group by a lattice.**Submetry**a short map "f" between metric spaces called submetry if for any point "x" and radius "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**. The "k"-systole of "M", $syst\_k(M)$, is the minimal volume of "k"-cycle nonhomologous to zero.**T**Tangent bundle **Totally convex.**A subset "K" of a Riemannian manifold "M" is called totally convex if for any two points in "K" any geodesic connecting them lies entirely in "K", see also "convex".**Totally geodesic**submanifold is a "submanifold" such that all "geodesics" in the submanifold are also geodesics of the surrounding manifold.**W**on a group is a metric of theWord metric Cayley graph constructed using a set of generators.

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