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': MM 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 Da, and a torsion tensor Tab.
* Riemann-Cartan manifold: A differentiable manifold with a metric gab 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; Katanaev a1311; 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 [Spinc 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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