Differentiable
Manifolds and Differential Topology| Differentiable
Manifolds and Differential Topology |
In General > s.a. diffeomorphisms; differentiable
maps; embedding;
manifolds.
* Differentiable structure:
An assignment of an equivalence class
of atlases, with charts related by differentiable transition functions.
* Differentiable manifold:
A space with a differentiable structure, or a topological
manifold
with
a
sheaf of k-smooth
functions (a ring space), or differentiable relations
between charts (some topological manifolds admit no such structures).
* Motivation: The differentiable
manifold structure is the most natural and general one in which to study differentiability;
Differential topology studies properties of differentiable manifolds without
additional structure (diffeos, forms, tensors and concomitants, etc);
They can be studied as ways of reducing a topological tangent bundle to
a tangent
bundle [@ Milnor Top(64)].
* History: The first
to present a coordinate-independent account was Darboux.
* Remark: One atlas is
sufficient to specify the differentiable structure,
and any C1 atlas is C1-equivalent
to a smooth one.
Algebraic Description > s.a. manifolds [Gel'fand-Naimark
theorem].
* Idea: The whole structure of a differentiable manifold can be recovered
just from its ring of differentiable functions, by defining it to be the set
of maximal ideals of this ring.
@ References: in Milnor & Stasheff 74; Yodzis PRIA(75).
Manifolds with Inequivalent Differentiable Structures > s.a.
4D manifolds.
* For Rn:
For n 
4
dimensions, Rn admits
a unique differentiable structure; R4 has
uncountably many inequivalent ones.
* For spheres: For Sn
with n 
6
the differentiable structure is unique; For higher n, the number N(n)
is
n:
7 8 9 10 11
N(n): 28 2 8 6 992 ;
The exotic ones are also boundaries of manifolds, but not of contractible
ones.
* General results: Any
closed, connected 2- or 3-manifold has a unique differentiable structure; In n 
4
dimensions, some manifolds don't admit any, others many inequivalent ones;
For n 5,
compact topological manifolds have a finite number of differentiable structures
[@
Kirby & Siebenmann
77], but the PL and Diff structures coincide for n up to 6.
@ References: Castelvecchi SA(09)aug [Kervaire problem, the classification of
exotic
higher-dimensional
spheres]
Examples and Types > s.a. 2D, 3D, 4D
manifolds; Contact
Manifold; spheres; Surfaces.
$ Torus: The n-dimensional
torus is the manifold T n 
Cn given
by T n:=
{z 
Cn |
|zi| = 1, i = 1, ..., n}.
$ Stiefel manifold of k-frames:
The manifold Vn, n–k(R)
= O(n)/O(n–k) of all k-frames in n-dimensional
space; Properties: 
p Vn, n–k(R)
= 0 for 0 p n–k–1.
> Generalizations: see differential
geometry; non-commutative geometry.
Related Topics and References > s.a. differential
forms; exterior
calculus; tensors.
@ Texts: Milnor 58, 65; Wallace 66; Hirsch 76; Bröcker & Jänich
82; Gauld 82; Guillemin & Pollack 84; Pontrjagin 86; Bredon 97.
@ With boundaries: Margalef-Roig & Outerelo Domínguez 92;
Scott & Szekeres
JGP(94)gq;
Fama & Clarke CQG(98)gq.
@ Higher-order tangent structures: Dodson & Radivoiovici IJTP(82);
> s.a. Jet Bundle.
> Other topics: see integration; Isotopy;
morse
theory; Stokes'
Theorem; tangent
structures.
Differentiable Structures and Physics > s.a. 4D
manifolds; singularities;
[quantum spacetime].
@ General references: Chamblin JGP(94)gq/95 [homotopy
type and causal structure]; Król FP(04)
[exotic R4
and non-commutative spaces]; Asselmeyer-Maluga & Brans 07; Brans GRG(08)
[rev].
@ And phase transitions in field theory: Goldin & Moschella JPA(95).
@ Exotic structures and gravity: Penrose in(84);
Witten CMP(85);
Rohm AP(89);
Brans & Randall GRG(93)gq/92;
Brans CQG(94)gq,
JMP(94)gq,
gq/96-in;
Asselmeyer CQG(97)gq/96;
Schleich & Witt CQG(99)gq [7D
Euclidean quantum gravity]; Sladkowski gq/99 [consequences];
Aßelmeyer-Maluga & Brans GRG(02)gq/01 [and
cosmology]; Boyer et al EM(05)m.DG/03 [Einstein
metrics on exotic S7, S11 and
S15]; Krol ht/05-in
[model theory approach]; Asselmeyer-Maluga & Rose gq/05 [geometrization
of quantum mechanics]; Sladkowski a0910-in [astrophysical consequences].
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oct 2009