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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.
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.
AEdit
Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2).
Arc-wise isometry the same as path isometry.
Autoparallel the same as totally geodesic.<ref>Template:Cite book</ref>
BEdit
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 : [0, ∞)→X, the Busemann function is defined by<math display="block">B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)-p|-t).</math>
CEdit
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 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.
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>Template:Cite journal</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.
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.
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.
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>Template:Citation</ref> Sometimes the additional requirement is made that the distance function to <math display="inline">p</math> in these balls is convex.<ref>Template:Citation</ref>
DEdit
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.
EEdit
Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry)
FEdit
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.
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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-hyperbolic metric space
Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.
HEdit
Hadamard space is a complete simply connected space with nonpositive curvature.
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)
IEdit
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>Template:Cite book</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 2r 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>Template:Citation</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>Template:Cite book</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>Template:Citation</ref>
JEdit
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>
KEdit
LEdit
Length metric the same as intrinsic metric.
Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.
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>Template:Cite book</ref>
Logarithmic map, or logarithm, is a right inverse of Exponential map.<ref>Template:Cite book</ref><ref>Template:Cite journal</ref>
MEdit
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.
NEdit
Natural parametrization is the parametrization by length.<ref>Template:Cite book</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>Template:Cite book</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.
OEdit
Orthonormal frame bundle is the bundle of bases of the tangent space that are orthonormal for the Riemannian metric.
PEdit
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.
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>Template:Citation</ref>
QEdit
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>Template:Citation</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.
REdit
Radius of metric space is the infimum of radii of metric balls which contain the space completely.<ref>Template:Cite book</ref>
Ray is a one side infinite geodesic which is minimizing on each interval.<ref>Template:Cite journal</ref>
Riemann The mathematician after whom Riemannian geometry is named.
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 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.
SEdit
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, Sp: TpM→TpM. 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.
Sol manifold is a factor of a connected solvable Lie group by a lattice.
Submetry A short map f between metric spaces is called a submetry<ref>Template:Cite journal</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.
TEdit
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>.
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>Template:Cite journal</ref>
Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.<ref>Template:Cite journal</ref>
UEdit
Uniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic.
VEdit
WEdit
Word metric on a group is a metric of the Cayley graph constructed using a set of generators.