词条 | Glossary of Riemannian and metric geometry |
释义 |
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:
Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or 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. {{compact ToC|side=yes|top=yes|num=yes}}AAlexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2) Almost flat manifoldArc-wise isometry the same as path isometry. Autoparallel the same as totally geodesicBBarycenter, see center of mass. bi-Lipschitz map. A map is called bi-Lipschitz if there are positive constants c and C such that for any x and y in XBusemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by == C == 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 extended Einstein's General relativity to Einstein–Cartan theory, using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides affine torsion, which allows for non-symmetric curvature tensors and the incorporation of spin-orbit coupling. Center of mass. A point q ∈ M is called the center of mass of the points if it is a point of global minimum of the function Such a point is unique if all distances are less than radius of convexity. Christoffel symbolCollapsing manifoldComplete spaceCompletionConformal map is a map which preserves angles. Conformally flat a 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 are called conjugate if there is a Jacobi field on which has a zero at p and q. Convex function. A function f on a Riemannian manifold is a convex if for any geodesic the function is convex. A function f is called -convex if for any geodesic with natural parameter , the function 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 bundleCovariant derivativeCut locusDDiameter of a metric space is the supremum of distances between pairs of points. Developable surface is a surface isometric to the plane. Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz. EExponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)FFinsler metricFirst fundamental form for an embedding or immersion is the pullback of the metric tensor. GGeodesic is a curve which locally minimizes distance. 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 where is a geodesic. Gromov-Hausdorff convergenceGeodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic. HHadamard space is a complete simply connected space with nonpositive curvature. Horosphere a level set of Busemann function. IInjectivity 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. 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. 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 on N. An orbit space of N by a discrete subgroup of which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold. Isometry is a map which preserves distances. Intrinsic metricJJacobi 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 with , then the Jacobi field is described by Jordan curveKKilling vector fieldLLength metric the same as intrinsic metric. Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds. 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). Lipschitz mapLogarithmic map is a right inverse of Exponential map. MMean curvatureMetric ballMetric tensorMinimal surface is a submanifold with (vector of) mean curvature zero. NNatural parametrization is the parametrization by length. Net. A sub set S of a metric space X is called -net if for any point in X there is a point in S on the distance . 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 -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 imbedding of a manifold M into an ambient Euclidean space , the normal bundle is a vector bundle whose fiber at each point p is the orthogonal complement (in ) of the tangent space . Nonexpanding map same as short mapPParallel transportPolyhedral 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. Path isometryProper 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. QQuasigeodesic has two meanings; here we give the most common. A map (where is a subsegment) is called a quasigeodesic if there are constants and such that for everyNote that a quasigeodesic is not necessarily a continuous curve. Quasi-isometry. A map is called a quasi-isometry if there are constants and such that 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. RRadius 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 interval Riemann curvature tensorRiemannian manifoldRiemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time. SSecond 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, It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space. 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 (there is no standard agreement whether to use + or − in the definition). Short map is a distance non increasing map. Smooth manifoldSol 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 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. Sub-Riemannian manifoldSystole. The k-systole of M, , is the minimal volume of k-cycle nonhomologous to zero. TTangent bundleTotally 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. UUniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic. WWord metric on a group is a metric of the Cayley graph constructed using a set of generators. {{DEFAULTSORT:Glossary Of Riemannian And Metric Geometry}} 3 : Glossaries of mathematics|Metric geometry|Riemannian geometry |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。