Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 11
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. . , x n , y 1 , . . . , y n , z) in which ω = dz −niii=1 y dx .contact manifold The odd-dimensional counterpart of a symplectic manifold. A contactmanifold is a smooth (2n+1)-dimensional manifold M together with a one-form ω on M suchthat ω∧(dω)n = 0 everywhere on M. There is aunique vector field V on M, called the characteristic vector field, determined by two conditions:(i.) ω(V ) = 1 and(ii.) dω(V , W ) = 0 for all W .In special local coordinates, this is the vector∂field ∂z.An example of a contact manifold is the unitsphere S 2n+1 , viewed as a submanifold of complex n-dimensional space. The characteristicvector field is the field of unit vectors tangentto the great circles which form the fibers of theHopf fibration π : S 2n+1 −→ CPn of the sphereover complex projective space.
π is the mapthat takes a point in the sphere to the complex 1dimensional subspace containing the point. The1-form ω is given by the formulaω(W ) =< V , W > ,where <, > is the Euclidean inner product.contact metric structure A Riemannian metric g on a manifold M of dimension 2n + 1 anda contact form η which are compatible. Thecontact form η is a 1-form on M which satisfiesthe condition η ∧ (dη)n = 0 at every point. ηdetermines a subspace K(p) of codimension 1in the tangent space Mp to M at the point p,as well as a vector ξ transverse to K(p) anddetermined by the conditions: η(ξ ) = 1 anddη(V , ξ ) = 0 for all V in K(p).
The metric g must make ξ(p) a unit vector, orthogonal to K(p). For X and Y in K(p), the metricsatisfies dη(X, Y ) = g(X, φY ), where φ(p) :Mp → Mp is a linear transformation satisfyingφ 2 (V ) = −V on K(p) and φ(ξ ) = 0. Moreprecisely, φ is a tensor field on M of type (1, 1)satisfying φ 2 = −I + η ⊗ ξ .© 2001 by CRC Press LLCcontact structure A specification of a (2n)dimensional plane K(p) in the tangent spaceof a manifold M of dimension 2n + 1, at eachpoint p, in such a way that in some open set Uaround p there is a smooth, one-form ω such thatK(q) is the kernel of ω(q) for every q.
The oneform is required to satisfy the non-degeneracycondition: ω ∧ (dω)n = 0 for all q in U . Theform ω is called a local contact form. On theoverlap of two such open sets, the local formsagree up to a non-vanishing scalar multiple. Ifthe form can be globally defined, then M is acontact manifold.An example of a contact structure is givenby the manifold of straight lines in the planeR2 .
Local coordinates (x, y, z) are given byletting (x, y) be the point of contact and 0 <z < π the angle between the line and a horizontal line. (Different coordinates are chosen ifthe line is horizontal.) The 1-form is given byω(x, y, z) = sin(z)dx − cos(z)dy. This 1-formcannot be defined globally.continued fractionforma0 +a1 +A real number of theb1b2a2 +b3b4a3 +a4 +...where each ai is a real number. If the expression consists of only a finite number of fractions,the expression is called a finite continued fraction and is (obviously) a rational number. Seealso finite continued fraction. An infinite simplecontinued fraction is one of the forma0 +1a1 +1a2 +1a3 +1...where each of the ai is an integer.
See alsoconvergence of a continued fraction.continuous functionA function f from atopological space X to a topological space Ysuch that, for each open U ⊆ Y , the set f −1 (U )is open in X. For functions on the real line, thisis equivalent to requiring that, for any > 0,there exists a δ > 0 such that |f (x) − f (y)| < whenever |x − y| < δ.convex closurecontinuous geometry An orthomodular lattice, i.e., a complete and complemented modular lattice L such that given any element x ofL and any subset W of L which is well ordered with respect to the ordering in L, thenx ∩ sup w = sup(a ∩ w), where w ∈ W .
Theconcept of a continuous geometry was introduced by John von Neumann. When the dimension is discrete, continuous geometry containsprojective geometry as a special case. Generally, however the lattices have continuous dimension.(the nth convergent of the continued fraction).If lim Cn = L, then the continued fraction iscontinuum hypothesisconvergent sequenceA sequence of points{xn : n ∈ N} in a topological space X convergesto x ∈ X if, for any open set U containing x,there is an N ∈ N with xn ∈ U for all n ≥ N .
Asequence is convergent if it converges to somex ∈ X. For example, the sequence of reals { n1 }converges to 0, while the sequence {n} does notconverge.2ℵ0The statement= ℵ1 .(It is not possible to prove this statement or itsnegation in Zermelo-Frankel set theory with theAxiom of Choice.)contractible topological spaceA topological space X that can be shrunk to a point.
Moreprecisely, X is contractible if there is a continuous function c : X ×[0, 1] −→ X (called a contraction) such that c(x, 0) = x for all x in X, andfor some p in X, c(x, 1) = p for all x. The ndimensional Euclidean spaces are contractible,while any space with a nonzero homology or homotopy group in a positive dimension gives anexample of a non-contractible space.n→∞said to converge to L. That is,L = a0 +b1a1 +b2a2 +a3 +b3b4a4 +convergentfraction....See convergence of a continuedconvex (1) A non-empty subset X of Rn suchthat, for any elements x, y ∈ X, and any numberc such that 0 ≤ c ≤ 1, the element cx +(1−c)yof Rn belongs to X.
In R2 , for example, a setis convex if it contains the line segment joiningany two of its points. See also convex closure.(2) A real function f (x) in an interval I suchthat the graph of f lies nowhere above its secantline in any subinterval of I .contractionIf X is a subspace of Y , then acontraction of X in Y is a continuous functionc : X × [0, 1] −→ Y such that c(x, 0) = x forall x in X, and for some p in Y , c(x, 1) = p forall x in X. See contractible topological space.convex body A bounded, closed, convex set(finite or infinite) that has interior points. Seeconvex.convergence of a continued fractionthe continued fractionconvex cellThe convex closure of a finiteset of points P = {p0 , p1 , ..., pk }, in an ndimensional affine space An .
When p0 , ..., pkare independent, it is a k-dimensional simplexwith vertices p0 , ..., pk .a0 +b1a1 +b2a2 +a3 +b3b4a4 +...defineCn = a0 +b1a1 +© 2001 by CRC Press LLCb2a2 +a3 +b3b4...bn−1an−1 + abnnGivenconvex closureFor any subset X of an ndimensional affine space An , there exists a minimal convex set that contains X. This set, whichis the intersection of all the convex sets that contain X, is the convex closure of X.
In Rn , theconvex closure of a set X is the set of possiblelocations of the center of gravity of mass whichcan be distributed in different ways in the minimal convex set containing X. Each point of theconvex coneconvex closure is the center of gravity of a massconcentrated at not more than n + 1 points.For a subset X of An , X is convex if the segment joining two arbitrary points of X is contained in X.convex coneA convex body consisting ofhalf-lines emanating from a point (the apex ofthe cone).
The surface of a convex cone is sometimes called a convex cone.convex cylinder A cylinder that lies entirelyon one side of any tangent plane at a point of thecylinder. See cylinder.convex hull The smallest convex set containing a given subset X of a Euclidean space. Theconvex hull of X can be constructed by formingthe intersection of all half-planes containing X.See also convex closure.convex polygonA polygon in R2 with theproperty that each of its interior angles (the angles made by adjacent sides of the polygon andcontained within the polygon) is less than orequal to 180 degrees. A convex polygon alwayshas an interior.convex polyhedral coneA convex cone inR3 which is the intersection of linear half-spaces.See convex cone.convex polyhedronThe convex closure ofa finite number of points in Rn ; that is, thebounded intersection of a finite number of closedhalf-spaces.
In R3 , it is a solid bounded by planepolygons, which lies entirely on one side of anyplane containing one of its faces. Any plane section of a convex polyhedron is a convex polygon.See convex closure, convex polygon.coordinateSee coordinate system. In R2 , acoordinate is one of an ordered pair of numbersthat locates the position of a point in the plane.coordinate axisOne of finitely many components of a reference system which providesa one-to-one correspondence between the elements of a set (on a plane or a surface, in aspace, or on a manifold) with the numbers usedto specify their position. In Rn , it is part of an© 2001 by CRC Press LLCorthogonal frame which determines the rectangular coordinates of each point in the space.
InR2 a coordinate axis is a line along which orparallel to which a coordinate is measured.coordinate bundle Let E, B, F be topological spaces and p : E → B be a continuous map.Let G be an effective left topological transformation group of F . If there exists an open covering {Uα }α∈ of B, and a homeomorphism φα :Uα × F ≈ p −1 (Uα ) for each α ∈ , then thesystem (E, p, B, F, G, Uα , φα ) is a coordinatebundle if it has the following three properties:(i.) pφα (b, y) = b (b ∈ Uα , y ∈ F ); (ii.) define φα,β : F ≈ p−1 (b) (b ∈ U α) by φα,β (y) =−1φα (b, y). Then gβα (b) = φβ,bφα,b ∈ G forb ∈ Uα ∩ Uβ ; (iii.)gβα (b) : Uα ∩ Uβ → G iscontinuous.coordinate functionThe homeomorphismφα : Uα × F ≈ p −1 (Uα )for each α ∈ of the coordinate bundle (E, p,B, F, G, Uα , φα ) belonging to the fiber bundle(E, p, B, F, G).coordinate hyperplane A coordinate hyperplane in a vector space X over a field K is theimage under a translation of a vector subspaceM with the quotient space X/M.coordinate neighborhoodThe open covering {Uα }(α ∈ ) of B of the coordinate bundle(E, p, B, F, G, Uα , φα ) belonging to the fiberbundle (E, p, B, F, G).
See coordinate bundle.coordinate systemLet S be a set of mathematical objects. A coordinate system is a mechanism that assigns (tuples of) numbers to eachelement of the set S. The numbers corresponding to each element are called its coordinates. InR2 , such a reference system is called the rectangular coordinate system.
In R3 , the coordinatefunctions, u and v, of p−1 [where p is a one-toone regular mapping of an open set of R2 intoa subset of R3 ] constitute the coordinate systemassociated with p.coordinate transformationLet E, B, F betopological spaces and p : E → B be a continuous map. Let G be an effective left topologicalcovariant functortransformation group of F .
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