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Riemannian and Metric Geometry Glossary

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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

A |  B  |  C |  D  |  E |  F  |  G |  H |  I |  J  |  K |  L |  M |  N  |  P |  Q  |  R |  S |  T |  W

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for transporting tangent vectors to a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields: the infinitesimal transport of a vector field in a given direction.

Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory. The local theory concerns itself primarily with notions of parallel transport and holonomy. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups. An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field. A Koszul connection is a connection generalizing the derivative in a vector bundle.

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually an Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold. This article deals primarily with the first concept.

The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.

In a plane, this is a scalar quantity, but in three or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.

The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.

Metric space

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

The geometric properties of the space depend on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.

A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point. The terms are named after German mathematician Bernhard Riemann.

A Riemannian metric makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

Riemannian manifolds should not be confused with Riemann surfaces, manifolds that locally are patches of the complex plane.

* * *

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.

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

In mathematics, a smooth compact manifold M is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon on M such that  \mbox{diam}(M,g_\varepsilon)\le 1 and  g_\varepsilon is \varepsilon-flat, i.e. for sectional curvature of  K_{g_\varepsilon} we have  |K_{g_\epsilon}| < \varepsilon.

In fact, given n, there is a positive number \varepsilon_n>0 such that if a n-dimensional manifold admits an \varepsilon_n-flat metric with diameter \le 1 then it is almost flat. On the other hand you can fix the bound of sectional curvature and finally you get the diameter going to zero, so the almost flat manifold is a special case of a collapsing manifold, which is collapsing along all directions.

According to the Gromov—Ruh theorem, M is almost flat if and only if it is infranil. In particular, it is a finite factor of a nilmanifold, which is the total space of a principal torus bundle over a principal torus bundle over ... over a torus.

Arc-wise isometry the same as path isometry.

B

Barycenter, see center of mass.

bi-Lipschitz map. A map f:X\to Y is called bi-Lipschitz if there are positive constants c and C such that for any x and y in X

c|xy|_X\le|f(x)f(y)|_Y\le C|xy|_X

Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by

B_\gamma(p)=\lim_{t\to\infty}(|\gamma(t)p|-t)

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 p_1,p_2,\dots,p_k if it is a point of global minimum of the function

f(x) = | pix | 2
i

Such a point is unique if all distances | pipj | are less than radius of convexity.

Christoffel symbol

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols. The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.

At each point of the underlying n-dimensional manifold, for any local coordinate system, the Christoffel symbol is an array with three dimensions: n × n × n. Each of the n3 components is a real number.

Under linear coordinate transformations on the manifold, it behaves like a tensor, but under general coordinate transformations, it does not. In many practical problems, most components of the Christoffel symbols are equal to zero, provided the coordinate system and the metric tensor possess some common symmetries.

In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor.

Collapsing manifold

In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gn, such that as n goes to infinity the manifold is close to a k-dimensional space, where k < n, in the Gromov–Hausdorff distance sense. Generally there are some restrictions on the sectional curvatures of (M, gn). The simplest example is a flat manifold, whose metric can be rescaled by 1/n, so that the manifold is close to a point, but its curvature remains 0 for all n.

Complete space

In mathematical analysis, a metric space M is called complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). Thus, a complete metric space is analogous to a closed set. For instance, the set of rational numbers is not complete, because e.g. \sqrt{2} is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.

Completion

For any metric space M, one can construct a complete metric space M' (which is also denoted as \scriptstyle\overline{M}), which contains M as a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f' from M' to N, which extends f. The space M' is determined up to isometry by this property, and is called the completion of M.

The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences (xn)n and (yn)n in M, we may define their distance as

d(x,y) = limn d(xn,yn).

Conformal 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 γ are called conjugate if there is a Jacobi field on γ which has a zero at p and q.

Convex function. A function f on a Riemannian manifold is a convex if for any geodesic γ the function f\circ\gamma is convex. A function f is called λ-convex if for any geodesic γ with natural parameter t, the function f\circ\gamma(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

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle.

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher dimensional Euclidean space, the covariant derivative can be viewed as the orthonormal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.

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 of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz.

E

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection. Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

F

Finsler metric

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve γ : [a,b] → M is given by the length functional

L[\gamma] = \int_a^b F(\gamma(t),\dot{\gamma}(t))\,dt,

where F(x, · ) is a Minkowski norm (or at least an asymmetric norm) on each tangent space TxM. Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor).

Élie Cartan (1933) named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation (Finsler 1918).

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

G

Geodesic 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 (γ(t),γ'(t)) where γ is a geodesic.

Gromov-Hausdorff convergence

In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of 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.

Horosphere a level set of Busemann function.

I

Injectivity 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.

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 of semidirect product N \rtimes F on N. A compact factor of N by subgroup of N \rtimes F acting freely on N is called infranil manifold. Infranil manifolds are factors of nil manifolds by finite group (but the converse fails).

Isometry is a map which preserves distances.

Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the length of all paths from one point to the other. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.

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 γτ with γ0 = γ, then the Jacobi field is described by

J(t)=\partial\gamma_\tau(t)/\partial \tau|_{\tau=0}.\,

Jordan curve

In mathematics, a curve (sometimes also called curved line) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional (plane curves) or three-dimensional (space curves) Euclidean space are of interest.

Different disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In every day language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields.

K

Killing vector field

In mathematics, a Killing vector field (often just Killing and abbreviated to KVF) , named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.

L

Length metric the same as intrinsic metric.

Levi-Civita connection is a natural way to differentiate vector field 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 map

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: For every pair of points on the graph of this function, their secant line-segment's slope has absolute-value no greater than a definite real number; this constant, which is the same for all the secants, is called the function's "Lipschitz constant" (or "modulus of uniform continuity").

In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem.

The concept of Lipschitz continuity is well-defined on metric spaces. A generalisation of Lipschitz continuity is called Hölder continuity.

Logarithmic map is a right inverse of Exponential map.

M

Mean curvature

In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.

The concept was introduced by Sophie Germain in her work on elasticity theory.

Metric ball

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.

A manifold equipped with a metric tensor is known as a Riemannian manifold. By integration, the metric tensor allows one to define and compute the length of curves on the manifold. The curve connecting two points that has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p,q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.

While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita who first codified the notion of a tensor. The metric tensor is an example of a tensor field, meaning that relative to a local coordinate system on the manifold, a metric tensor takes on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.

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

N

Natural parametrization is the parametrization by length.

Net. A sub set S of a metric space X is called ε-net if for any point in X there is a point in S on the distance \le\epsilon. This is distinct from topological nets which generalise limits.

Nil manifolds: the minimal set of manifolds which includes a point, and has the following property: any oriented S1-bundle over a nil manifold is a nil manifold. 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 {\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 TpM.

Nonexpanding map same as short map

P

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection. Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.

The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, vice versa, parallel transport is the local realization of a connection.

As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose-Singer theorem makes explicit this relationship between curvature and holonomy.

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.

Path isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.

Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

An isometric surjective linear operator on a Hilbert space is called a unitary operator.

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: \textbf{R} \to Y is called quasigeodesic if there are constants K > 0 and C \ge 0 such that

{1\over 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 f:X\to Y is called a quasi-isometry if there are constants K \ge 1 and C \ge 0 such that

{1\over K}d(x,y)-C\le 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 interval

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point. The terms are named after German mathematician Bernhard Riemann.

A Riemannian metric makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

Riemannian manifolds should not be confused with Riemann surfaces, manifolds that locally are patches of the complex plane.

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

S

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,

II(v,w)=\langle S(v),w\rangle

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, SpTpMTpM. If n is a unit normal field to M and v is a tangent vector then

S(v)=\pm \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

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. Note that a differentiable manifold as it stands does not have any metric structure or any notion of orthogonality. The addition of metric (or pseudo-metric) structure corresponds to the linear space mentioned above actually being Euclidean space (or pseudo-Euclidean space).

More formally, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called transition maps.

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 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

In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces.

Sub-Riemannian manifolds (and so, a fortiori, Riemannian manifolds) carry a natural intrinsic metric called the metric of Carnot–Carathéodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold).

Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure.

Systole. The k-systole of M, systk(M), is the minimal volume of k-cycle nonhomologous to zero.

T

Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M. That is,

TM = \bigsqcup_{x\in M}T_xM=\bigcup_{x\in M} \left\{x\right\}\times T_xM.

where TxM denotes the tangent space to M at the point x. So an element of TM can be thought of as a pair (xv), where x is a point in M and v is a tangent vector to M at x. There is a natural projection

 \pi : TM \twoheadrightarrow M

defined by π(xv) = x. This projection maps each tangent space TxM to the single point x.

The tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of TM is a vector field on M, and the dual bundle to TM is the cotangent bundle, which is the disjoint union of the cotangent spaces of M. By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum TME is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n=1,3,7 (by results of Bott-Milnor and Kervaire).

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

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






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Published - May 2011







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