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Whitney Immersion Theorem

Whitney immersion theorem

In differential topology, the Whitney immersion theorem states that for

m>1

, any smooth second-countable

m

-dimensional manifold can be immersed in Euclidean

2m-1

-space.

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using calculus. These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical systems.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one (see the Nash embedding theorem).

Technical requirements

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature. A differential manifold is a topological space with a collection of homeomorphisms from open sets of the space to open subsets in Rⁿ such that the open sets cover the space, and if f, g are homeomorphisms then the function f o g ^-1 from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every homeomorphism results in an infinitely differentiable function from the open unit ball to R. At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rⁿ. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability. A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point. An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^- of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

Differential topology

Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). Smooth manifolds are 'softer' than manifolds with extra geometric stuctures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume. Conversely, smooth manifolds are more rigid than the topological manifolds. Certain topological manifolds have no smooth structures at all (see Donaldson's theorem) and others have more than one inequivalent smooth structure (such as exotic spheres). Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.

Branches of differential geometry

Contact geometry

Contact geometry is an analog of symplectic geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form

\alpha

such that

\alpha\wedge (d\alpha)^n

does not vanish anywhere.

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is much more general structure than a Riemannian metric.

Riemannian geometry

Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space. These allow one to generalise the notion from Euclidean geometry and analysis such as gradient of a function, divergence, length of curves and so on; without assumptions that the space is globally so symmetric. The Riemannian curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.

Symplectic topology

Symplectic topology is the study of symplectic manifolds, which can occur only in even dimensions. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form). Unlike in Riemannian geometry, all symplectic manifolds are locally isomorphic, so the only invariants of a symplectic manifold are global in nature.

External links

- [http://rsp.math.brandeis.edu/3D-XplorMath/Surface/a/bk/curves_surfaces_palais.pdf A Modern Course on Curves and Surface, Richard S Palais, 2003]
- [http://rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery.html Richard Palais's 3DXM Surfaces Gallery]
- [http://www.cs.elte.hu/geometry/csikos/dif/dif.html Balázs Csikós's Notes on Differential Geometry]

Reference books

1. A Comprehensive Introduction to Differential Geometry (5 Volumes), 3rd Edition by Michael Spivak (1999) 2. Differential Geometry of Curves and Surfaces by Manfredo Do Carmo (1976). A classical geometric approach to differential geometry without the tensor machinery. 3. Riemannian Geometry by Manfredo Perdigao do Carmo, Francis Flaherty (1994) 4. Geometry from a Differentiable Viewpoint by John McCleary (1994) 5. A First Course in Geometric Topology and Differential Geometry by Ethan D. Bloch (1996) 6. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. by Alfred Gray (1998) ja:微分幾何学

Embedding

:For other uses of this term, see embedded (disambiguation). In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.

Topology/Geometry

General topology

In general topology, an embedding is a homeomorphism onto its image. More explicitly, a map f : X → Y between topological spaces X and Y is an embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : X → Y lets us treat X as a subspace of Y. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image f(X) is neither an open set nor a closed set in Y.

Differential geometry

In differential geometry: Let M and N be smooth manifolds and

f:M\to N

be a smooth map, it is called an immersion if for any point

x\in M

the differential

d_xf:T_x(M)\to T_(N)

is injective (here

T_x(M)

denotes tangent space of

M

x

). Then an embedding, or a smooth embedding, is defined to be an immersion which is an embedding in the above sense (i.e. homeomorphism onto its image). When the manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion. In other words, an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is a local embedding (i.e. for any point

x\in M

there is a neighborhood

x\in U\subset M

such that

f:U\to N

is an embedding.) An important case is N=Rⁿ. The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding. The less restrictive condition of immersion applies to the Boy's surface—which has self-intersections.

Riemannian geometry

In Riemannian geometry: Let (M,g) and (N,h) be Riemannian manifolds. An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i.e. g = f
- h. Explicitly, for any two tangent vectors :

v,w\in T_x(M)

we have :

g(v,w)=h(df(v),df(w))

. Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics. Equivalently, an isometric embedding (immersion) is a smooth embedding (immersion) which preserves length of curves (cf. Nash embedding theorem).

Algebra

Field theory

In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F. The kernel of σ is an ideal of E which cannot be the whole field E, because of the condition σ(1)=1. Therefore the kernel is 0 and thus any embedding of fields is a monomorphism. Moreover, E is isomorphic to the subfield σ(E) of F. This justifies the name embedding for an arbitrary homomorphism of fields.

Domain theory

In domain theory, an embedding of partial orders is

F

in the function space [X →Y] such that #

\forall x_1,x_2\in X: x_1\leq x_2\Leftrightarrow F(x_1)\leq F(x_2)

and #

\forall y\in Y:\

is directed. Based on an article from FOLDOC, used by permission.

Whitney embedding theorem

In differential topology, the Whitney embedding theorem states that any smooth second-countable

m

-dimensional manifold can be embedded in Euclidean

2m

-space. The result is sharp, in particular the projective

m

-space cannot be embedded into the Euclidean (

2m-1

)-space.

A little about the proof

Cases

m=1

and

2

can be done by hand. For

m\ge 3

a general position argument show that there is an immersion

f:M\to\mathbb R^

with transversal self-intersections. Then apply the Whitney trick, i.e. the following procedure which removes self-intersections one by one.

Whitney trick

Suppose

p\in\mathbb R^

is a point of self-intersection and

x,y\in M

such that

f(x)=f(y)=p

. Connect

x

and

y

by a smooth curve :

c:[0,1]\to M

so that

f\circ c

is a simple closed curve in

\mathbb R^

. Construct an embedding of a

2

-disc

h:D^2\to\mathbb R^

with boundary

f\circ c

. By a general position argument it can be constructed with no self-intersections and with no intersections with

f(M)

(here we use that

m\ge 3

). Then one can deform

f

in a little neighborhood of

h(D^2)

so that the self-intersection disappears. The last statement is very easy to see once you visualize this picture properly.

Other things coming from Whitney trick

The Whitney trick is used to prove h-cobordism theorem; it also shows that two oriented submanifolds of complimentary dimensions in a simply connected manifold of dimension

\ge 5

are isotopic to submanifolds such that all points of intersections have the same sign.

History

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept (which had been implicit in Riemann's work, Lie group theory, and general relativity for many years), building on Hermann Weyl's book The Idea of a Riemann surface.

Category:Differential topology

Category:Topology Category:Differential geometry

Choi (Korean name)

Choi, sometimes spelled Choe or Chey, is a common Korean family name. The 2000 South Korean census counted 2,169,704 people bearing this name. Many others live in North Korea and around the world. The vowel sound is similar to the German ö , but in English-speaking countries, most Chois prefer the anglicized form that rhymes with soy. Ethnic Koreans in former USSR prefer the form Tsoi (Tsoy) (Russian Цой), e.g. Russian rock artist Viktor Tsoi. There are roughly 160 clans of Chois. Most of these are quite small. The largest by far is the Gyeongju Choi clan, with a 2000 South Korean population of 976,820. The Gyeongju Choe claim the Silla scholar Choe Chi-won as their founder. Choi is also a common family name found in Chinese. Roughly 1% of the Chinese population bear this name. It is fabled that the origins of this name were derived from the intermixing of Koreans and Chinese eons ago.

External links

- [http://www.kj-choi.com/ Official site of the Gyeongju Choi Association, in Korean] Category:Korean family names

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Whitney immersion theorem

See also

Differential topology

Intrinsic versus extrinsic

Technical requirements

Differential topology

Branches of differential geometry

Contact geometry

Finsler geometry

Riemannian geometry

Symplectic topology

See also

External links

Reference books

Embedding

Topology/Geometry

General topology

Differential geometry

Riemannian geometry

Algebra

Field theory

Domain theory

See also

Whitney embedding theorem

A little about the proof

Whitney trick

Other things coming from Whitney trick

History

See also

Category:Differential topology

Choi (Korean name)

See also

External links