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 Γ^{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 Γ^{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.

* __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 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.

@ __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 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.

@ __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].

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

@ __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].

**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; 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.

@ __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 supersymmetric physics]; 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 fuzzy spheres]; Sykora a1610 ["construction kit"]; > 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].

@ __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]; Kunzinger JMAA(04)math/03 [non-smooth];
Jumarie PLA(07)
[fractional order]; 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 discrete
geometry; fractal; minkowski
space;
modified lorentz symmetry; quantum
group; quantum
spacetime; stochastic
processes.

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