Differential
Geometry| Differential
Geometry |
In General, Differentiable Geometric Structure > s.a. differentiable
manifolds.
* Idea: Differential
geometry studies properties of differentiable manifolds where a connection,
a metric, and/or some other geometrical
structure,
in addition to the differentiable
one, has been defined in terms of suitable tensor fields.
* Specification: A geometric
structure is usually specified by the number and kinds of fields one considers
on a differentiable manifold; In some cases however one can be specified in
a different way; For example, a metric geometry can be specified by an embedding
of the
manifold
in a higher-dimensional one, or by a sufficient number of axioms or integral
conditions and/or symmetries and constants.
* History: Started in
1864 by Christoffel; Developed by Ricci & Levi-Civita
in 1901.
* Rigidity: A geometrical structure is rigid of order n if, given
any two isometries 
,
': M →
M of it which agree, together with their first n derivatives,
at a point of M, = '.
* Affine manifold: A
differentiable manifold M with a linear connection 
abc,
defining a covariant derivative Da,
and a torsion tensor Tab.
* Riemann-Cartan manifold:
A differentiable manifold with a metric gab and
a metric-compatible connection abc (i.e.,
Da gbc =
0); > s.a. Riemann-Cartan.
@ Affine manifold: Nomizu & Sasaki 94; > s.a. affine
structure, torsion.
> Types: see complex
structure; connection [including affine
connection]; Contact, Frobenius,
Mirror Manifold; form [volume]; metric [including lorentzian
geometry and riemannian geometry]; spherical
symmetry; symplectic
geometry; tetrad [or more general frame/vielbein].
Other Concepts and Processes on Manifolds > s.a. loops;
stochastic processes.
* Isometry: A diffeomorphism f on
a manifold X that
leaves the metric g invariant, i.e., f *g = g;
For a manifold with non-degenerate metric
the isometry group is always finite-dimensional [@ Ashtekar & Magnon
JMP(78)].
* Geodesic completeness:
A manifold is geodesically complete if it has a complete affine connection;
Any compact Riemannian manifold is geodesically
complete,
but not all compact Lorentzian ones are; > s.a. Hopf-Rinow
Theorem.
@ Geodesic completeness: Kundt ZP(63)
[spacetime]; Misner JMP(63);
Meneghini m.CV/01-PhD,
m.CV/04 [for
complex geometry].
> Related concepts:
see curvature; diffeomorphisms; lie
groups.
General References > s.a. group
action; lie algebra; manifolds [supermanifolds,
fermionic degrees of freedom]; Willmore.
@ By physicists: Misner in(64); in Hawking & Ellis 73; in Misner
et al 73; Schmidt in(73); in Thirring 78; Eguchi et al PRP(80).
@ Books, II (mostly curves and surfaces): O'Neill 66; Millman & Parker
77; Bloch 96; Toponogov & Rovenski 05.
@ Books, III: Eisenhart 26, 47; Schouten 54; Lang 62; Flanders 63; Souriau
64; Sternberg 64; Bishop & Crittenden 64; Hicks 65; Pogorelov
67; Kobayashi & Nomizu 69; Brickell & Clark 70; Lang 72; Spivak 75;
Auslander & MacKenzie
77; Bishop & Goldberg 80; Klingenberg 82; O'Neill 83;
Dubrovin etal 85; Boothby 86; Abraham et al 88; D Martin 91; Chang 93; Kolár
et al 93 [unusual approach]; Chavel 94; Lang 95; Sharpe 97 [intro]; You & You
97 [including algebraic topology, non-commutative geometry, ...]; Chern et
al 99 [including Riemannian and Finsler].
@ Related topics: Yano 70 [integral formulas]; Kobayashi 72 [transformation
groups]; Hirsch 76 [infinite-dimensional];
Michor 80 [manifold of mappings]; Henderson & Taimina 98 [geometric intro].
And Physics > s.a. quantum field
theory.
@ Books: Schrödinger 63; Hermann 68; Geroch notes; Schutz 80;
Choquet-Bruhat et al 82 [III]; Trautman 84; Burke 85; Crampin & Pirani
86; Göckeler & Schücker 87; Visconti 87; in Arnold 89; Chau & Nahm
90; de Felice & Clarke 90; Frankel 97; Isham
99; Nakahara 03; Fecko 06.
@ Related topics: Kamien RMP(02)
[and soft matter]; Romero & Dahia m.DG/05/RBHM
[influence of general relativity on differential geometry]; Mallios IJTP(06)
[abstract differential geometry].
Generalizations > s.a. conformal
structures; differentiable manifolds [including
exotic]; geometry; manifold; metrics.
* Fuzzy manifold: A (non-commutative)
matrix model approximating the algebra of functions on a manifold.
@ Spectral point of view: Connes LMP(95); > s.a.
non-commutative geometry.
@ Quantum / deformed spaces: Kokarev mp/02-in;
Wachter EPJC(04)ht/02 [integration];
Bauer & Wachter
EJPC(03)mp/02 [operators].
@ Fuzzy / matrix approximations: Dolan & Nash JHEP(02)ht [Spinc structures];
Lizzi et al JHEP(03)
[fuzzy disk]; Balachandran et al ht/05-ln
[including supersymmetric physics]; > s.a. bessel
functions, Orbifold, topology
change.
@ Other ones: Giordano m.DG/03 [with
nilpotent infinitesimals]; Kunzinger JMAA(04)m.FA/03 [non-smooth];
Schuller & Wohlfarth NPB(06)ht/05,
JHEP(06)ht/05 [area
metric, stringy gravity]; Jumarie PLA(07)
[of fractional order].
> Related topics:
see discrete
geometry; fractal; minkowski
space;
modified lorentz symmetry; quantum
group; quantum
spacetime; stochastic
processes.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
26 jun 2008