Manifolds| Manifolds |
In General
$ Def: A topological
space in which every point has an open neighborhood homeomorphic to an open n-ball.
* Properties: It is finite-dimensional,
locally contractible, satisfies the Alexander-Lefschetz duality relationships.
Algebraic Characterization
* Idea: The structure
of a manifold can be recovered from the C*-algebra generated by appropriate
functions (abstractly, in
several
ways, e.g., as ideals or as propagators corresponding to point sources).
$ Gel'fand-Naimark theorem:
A C*-algebra 
with
identity is isomorphic to the C*-algebra of continuous bounded functions
on a compact Hausdorff space,
the spectrum
of ; The spectrum
can be constructed directly, as the set of maximal ideals, or *-homomorphisms → C.
$ Gel'fand-Kolmogorov theorem: (1939)
A compact Hausdorff topological space X can be canonically embedded
into the infinite-dimensional vector space C(X)*, the dual space
of the algebra of continuous functions C(X) as an "algebraic
variety" specified
by an infinite system of quadratic equations.
@ General references: Fell & Doran 88; Khudaverdian & Voronov a0709-in
[generalization
of Gel'fand-Kolmogorov].
@ Related topics:
Connes a0810 [spectral
triples]; > s.a. non-commutative geometry.
Additional Structures and Types of Manifolds > s.a. cell
complex; differentiable;
fiber bundles;
types of manifolds.
* Possibilities: Different levels of structure are manifold, triangulable
manifold, PL manifold, differentiable manifold; For a while, the first two
or three of these structures were conjectured to be equivalent, but
now this has been shown to be false.
* Multiplication structure: A map *: M × M → M.
* Comultiplication: A mapping F: C(M) → C(M)
C(M) on the
algebra of functions on M.
* Relationship: If M has a multiplication, then it gets a (diagonal)
comultiplication 
defined
by f 
(f):
f(x,y)
= F(x * y).
@ References: Kankaanrinta T&A(05)
[G-manifolds and Riemannian metrics].
Constructions and Operations on Manifolds > s.a. category;
embeddings;
tensor [product].
$ Direct product: Given two manifolds Xn and
Y p, their direct product
is Zn+p = Xn × Y p as
a set, with the product topology and the product
charts: UZ = UX × UY,
Z(x,y) =
(X(x), Y(y)).
$ Submanifold: N is a submanifold of M if it is a topological
subspace of M and the inclusion map is an embedding (if it is an immersion
we have
an immersed submanifold).
$ Integral submanifold:
A submanifold N 
M such
that for all p in N,
f*(TpN)
= Sp, with f : N → M the
embedding map.
$ Connected sum: In sloppy
notation, X # Y:= (X \
Dn) 
(Y \
Dn), where n is the dimension
of X and Y;
It is associative and commutative, and has Sn as
identity; Examples: X # Rn
= X \ {p}; > s.a. laplacian; 3D
manifolds.
@ Submanifolds: Carter JGP(92)
[outer curvature]; Giachetta et al mp/06 [Lagrangian
and Hamiltonian dynamics].
Superspace, Supermanifolds > s.a. complex
structures; Gegenbauer, Hermite and Jack
Polynomials.
* Idea: A manifold
with the bundle-like addition of a vector space of Grassmann numbers at each
point.
* Q-manifold: A supermanifold
equipped with an odd vector field satisfying {Q, Q} = 0.
@ General references: Rabin & Crane CMP(86);
Bagger pr(87); Nelson IJMPA(88);
Bandyopadhyay & Ghosh
IJMPA(89);
De Bie & Sommen AP(07)a0707 [Clifford
analysis approach]; Santi a0905 [homogeneous
supermanifolds]; Hubsch a0906 [size and algebro-geometric structure].
@ Texts and reviews: Hermann 77; DeWitt 92; Howe & Hartwell CQG(95);
Howe & Rogers 01; Sardanashvily a0910-ln.
@ Quantum superspace: Brink & Schwarz PLB(81).
@ Integration: Gates ht/97-in;
Cartier et al mp/02-in;
De Bie & Sommen JPA(07)-a0705 [and spherical harmonics]; > s.a. grassmann.
@ Fedosov supermanifolds: Geyer & Lavrov IJMPA(04)ht/03 [symplectic];
Lavrov
& Radchenko TMP(06);
Asorey et al a0809-in
[and Riemannian]; Monterde et al JGP(09).
@ Physics: Kochan JGP(04)m.DG/03 [supergeometry,
electromagnetism and gravity]; > s.a. scalar
fields [on
supersphere].
@ Related topics: Bruzzo & Pestov JGP(99);
Schwarz
LMP(00)ht [Q-manifolds
and gauge theory]; Philbin ht/03-wd
[topology]; Constantinescu JPA(05)ht/04,
MPLA(05)ht/04 [inner
product, Hilbert-Krein
structure]; Desrosiers et al m.CO/04 [symmetric
functions]; Sardanashvily IJGMP(08)
[supermetrics]; Asorey & Lavrov JMP(09) [symplectic and metric structures]; > s.a. brownian
motion; spherical
functions.
Other Types and Generalizations > see differential geometry [including fuzzy]; types of manifolds [including pseudomanifolds].
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oct
2009