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.
> Types of manifolds: see
cell complex; differential
geometry; differentiable manifolds;
types of manifolds [combinatorial, PL, topological, etc].
> Online resources: see the Manifold
Atlas Project site.
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 \({\cal A}\) with identity is isomorphic to the C*-algebra of continuous
bounded functions on a compact Hausdorff space, the spectrum of \({\cal A}\);
The spectrum can be constructed directly, as the set of maximal ideals, or *-homomorphisms
\({\cal A}\) → \({\mathbb C}\).
$ 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 AIP(07)-a0709 [generalization of Gel'fand-Kolmogorov];
Izzo AM(11).
@ Related topics: Connes a0810 [spectral triples];
> s.a. non-commutative geometry.
Additional Structures of Manifolds > s.a. fiber bundles.
* 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 Δ defined
by f \(\mapsto\) Δ(f): Δf(x,y)
= F(x * y).
@ References: Kankaanrinta T&A(05)
[G-manifolds and Riemannian metrics].
Constructions and Operations on Manifolds
> s.a. category; embeddings;
foliations; 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,
φZ(x,y) =
(φX(x),
φY(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 # \(\mathbb R\)n
= 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; geometric quantization.
* Idea: A manifold with the
bundle-like addition of a vector space of Grassmann numbers at each point.
* Q-manifolds: A Q-manifold
is a supermanifold equipped with an odd vector field satisfying {Q, åQ} = 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];
Hübsch a0906 [size and algebro-geometric structure];
Cattaneo & Schätz RVMP(11)-a1011-ln;
Voronov JGP(17)-a1409 [extending the category];
Castellani et al NPB(15)-a1503,
NPB(15)-a1507 [geometry, Hodge dual].
@ Texts, reviews: Hermann 77;
Chow IJTP(78);
DeWitt 92;
Howe & Hartwell CQG(95);
Howe & Rogers 01; Rogers 07;
Sardanashvily a0910-ln;
Fioresi & Lledó 15;
Hélein a2006-ln [intro].
@ Quantum superspace:
Brink & Schwarz PLB(81).
@ With metric structure:
Sardanashvily IJGMP(08) [supermetrics];
Asorey & Lavrov JMP(09) [symplectic and metric structures];
Dumitrescu et al JHEP(12)-a1205 [curved];
> s.a. killing tensors.
@ Integration: Gates ht/97-conf;
Cartier et al mp/02-in;
De Bie & Sommen JPA(07)-a0705 [and spherical harmonics];
> s.a. grassmann structures.
@ Fedosov supermanifolds: Geyer & Lavrov IJMPA(04)ht/03 [symplectic];
Lavrov & Radchenko TMP(06);
Asorey et al a0809-conf [and Riemannian];
Monterde et al JGP(09).
@ Q-manifolds: Schwarz LMP(00)ht [and gauge theory];
> s.a. characteristic classes.
@ Physics: Kochan JGP(04)m.DG/03 [supergeometry, electromagnetism and gravity];
Cirilo-Lombardo EPJC(12)-a1205 [geometrical properties];
Hack et al CMP(16)-a1501 [and locally covariant quantum field theory];
Nicolis a1606-conf
[and the partition function, fluctuations of a non-relativistic particle];
> s.a. Double Field Theory; FLRW
spacetimes; scalar fields [on supersphere].
@ Related topics: Bruzzo & Pestov JGP(99);
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];
Bruce ArchM-a1401 [curve and jet of a curve];
Randall a1412-th [closed forms in superspace];
Bruce & Ibarguengoytia a1806 [generalization, \(\mathbb{Z}_2^n\)-manifolds];
Klinker CMP(05)-a2001 [supergeometry and supersymmetry];
Bruce & Grabowski a2005 [odd connections];
> s.a. brownian motion; spherical functions.
> Other generalizations:
see differential geometry [including fuzzy]; differentiable
manifolds [exotic structures]; types of manifolds [including pseudomanifolds];
Wilson Loop.
main page
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 jun 2020