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 *ψ*, *ψ*':
*M* → *M* of it which agree, together with their first *n*
derivatives, at a point of *M*, *ψ* = *ψ*'.

* __Affine manifold__: A differentiable
manifold *M* with a linear connection \(\Gamma^a{}^~_{bc}\), defining
a covariant derivative *D*_{a},
and a torsion tensor *T*_{ab}.

* __Riemann-Cartan manifold__:
A differentiable manifold with a metric *g*_{ab}
and a metric-compatible connection \(\Gamma^a{}^~_{bc}\) (i.e., \(D^~_a g_{bc} = 0\);
> 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.

* __Isometries__: An isometry
on a manifold with metric (*X*, *g*) is a diffeomorphism
*f* 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.

@ __Isometries__: Frodden & Krasnov a2002 [in terms of the spin connection].

@ __Geodesic completeness__:
Kundt ZP(63) [spacetime];
Misner JMP(63);
Meneghini PhD-math/01,
CVEE(04)math [for complex geometry];
Sämann & Steinbauer in-a1310 [generalized spacetimes].

> __Related concepts__:
see curvature; diffeomorphisms;
lie groups; Surfaces [and singularities].

**General References**
> s.a. group action; lie algebra;
manifolds [supermanifolds, fermionic degrees of freedom];
Willmore Surfaces.

@ __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 06;
Woodward & Bolton 19.

@ __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 et al 85;
Boothby 86;
Abraham et al 88;
Martin 91;
Kolář et al 93 [unusual approach];
Chavel 94;
Lang 95; Sharpe 97 [intro];
Chern et al 99 [including Riemannian and Finsler];
Nicolaescu 07;
Taubes 11;
Borceux 14.

@ __Related topics__: Yano 70 [integral formulas];
Kobayashi 72 [transformation groups];
Hirsch 76 [infinite-dimensional];
Michor 80 [manifold of mappings];
Henderson & Taimina 98 [geometric intro];
Bielawski et al 11 [variational problems];
Snygg 12 [Clifford algebra approach];
> s.a. geometry [symbolic computational tools].

> __Online resources__: see Differential Geometry
Library site.

**And Physics**
> s.a. quantum field theory; riemann tensor.

@ __Books__: Schrödinger 63;
Hermann 68;
Geroch ln;
Schutz 80;
Choquet-Bruhat et al 82 [III];
Trautman 84;
Burke 85;
Crampin & Pirani 86;
Göckeler & Schücker 87;
in Arnold 89;
Chau & Nahm 90;
de Felice & Clarke 90;
Visconti 92;
Hou & Hou 97 [including algebraic topology, non-commutative geometry, ...];
Isham 99;
Rong & Yue 99;
Wang & Chen 99;
Nakahara 03;
Fecko 06;
Frankel 11;
Vargas 14 [including Clifford algebra, emphasis on forms];
Chakraborty a1908-ln.

@ __General references__:
Kamien RMP(02) [and soft matter];
Romero & Dahia RBHM-math/05 [influence of general relativity on differential geometry];
Mallios IJTP(06) [abstract differential geometry];
Chen IJGMP(13) [and quantum field theory];
> s.a. condensed matter [continuum mechanics].

@ __Dynamical systems__:
Burns & Gidea 05;
Ginoux 09.

**Generalizations**
> s.a. conformal structures; differentiable manifolds
[including exotic]; discrete geometry; geometry;
manifold; metrics.

* __Fuzzy manifold__: A (non-commutative)
matrix model approximating the algebra of functions on a manifold; These spaces have
received attention since they appeared as objects in string theory.

* __Synthetic Differential Geometry__: A
categorical generalization of differential geometry based on enriching the real line
with infinitesimals and weakening of classical logic to intuitionistic logic.

@ __General references__: Kunzinger JMAA(04)math/03 [non-smooth];
Jumarie PLA(07) [fractional order];
Nigsch & Vickers a1910 [distributional].

@ __Spectral point of view__: Connes LMP(95);
> s.a. non-commutative geometry.

@ __Quantum / deformed spaces__:
Kokarev in(04)mp/02;
Wachter EPJC(04)ht/02 [integration];
Bauer & Wachter EJPC(03)mp/02 [operators];
Goswami CMP(09) [quantum group of isometries].

@ __Fuzzy manifolds / matrix approximations__:
Dolan & Nash JHEP(02)ht [Spin^{c} structures];
Lizzi et al JHEP(03) [fuzzy disk];
Balachandran et al ht/05-ln [including supersymmetry];
Wang & Wang a1007 [area and dimension];
Govindarajan et al a1204-proc [phase structures of quantum field theories];
Mayburov PPN(12)-a1205 [quantum geometry and massive particles];
D'Andrea et al LMP(13)-a1209 [fuzzy sphere];
Chaney et al PRD(15)-a1506 [Lorentzian];
Sykora a1610 ["construction kit"];
Burić et al a1709 [fuzzy de Sitter space];
> s.a. bessel functions; non-commutative geometry
[spheres]; Orbifold; topology change.

@ __Area-metric spacetimes__: Schuller & Wohlfarth NPB(06)ht/05,
JHEP(06)ht/05 [stringy gravity];
Schuller et al AP(10)-a0908 [causal structure and classification];
Dahl IJGMP(12) [classification];
> s.a. cosmology in modified gravity;

cosmological acceleration in modified gravity;
gravitational lensing; relativistic particles.

@ __Synthetic Differential Geometry__: Heller & Król a1605 [and infinitesimal curvature],
a1607 [singularity problem and intuitionistic logic];
a1706 [and gravity].

@ __Other ones__: Giordano math/03 [with nilpotent infinitesimals];
Calin & Chang 09 [sub-Riemannian geometry];
Sardanashvily a0910 [in algebraic terms of modules and rings];
Balankin PLA(13) [scale-dependent spacetime metrics].

> __Related topics__:
see fractal; minkowski space;
modified lorentz symmetry; quantum group;
quantum spacetime; stochastic processes;
Sub-Riemannian Geometry.

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