Geometry

In General, Types of Geometries
* Idea: Different geometries can be related to each other by starting from the classical Euclidean geometry, and stating the sense in which each one generalizes one of its elements; If Euclidean geometry is X = \(\mathbb R\)ⁿ with the Euclidean group of rigid motions (SO(n) ×_s Tⁿ, rotations and translations) as symmetry group G, and has a globally defined positive-definite bilinear form on TX × TX, then the following are some of the generalizations.
* Riemannian geometry: Generalize the Euclidean bilinear form to a local metric (allow curvature); The local symmetry group is SO(n).
* Minkowskian geometry: Use a bilinear form with one negative eigenvalue; The symmetry group then becomes the Poincaré group; Can also allow more negative eigenvalue, to obtain other pseudo-Euclidean geometries).
* Lorentzian geometry: Generalize the Minkowskian bilinear form to a local metric (allow curvature); The local symmetry group is SO(1, n−1).
* Klein geometry: Generalize the global symmetry group G; One can express X = G/H, with H the stabilizer of some arbitrary x ∈ X.
* Cartan geometry: Generalize TX to another space that approximates X locally in Euclidean geometry; Or allow curvature in Klein geometry.
@ General references: Hilbert & Cohn-Vossen 52; Von Neumann 60; Blumenthal 61; Blumenthal & Menger 70; Dubrovin, Novikov & Fomenko 79; Rees 83; Ryan 86; Brannan et al 99 [II]; Reid & Szendrői 05 [and topology, II]; Benz 07 [classical geometries]; Peterson Sigma(07)-a0708 [directions of research]; Berger 10 [topics]; Borceux 14 [axiomatic]; Kappraff 14 [I, for non-mathematics students]; Johnson 18 [geometries and transformations]; Anderson a1811 [Brackets Consistency as a pillar of geometry].
@ Diophantine: Lang 60; Hindry & Silverman 00 [intro].
@ Other types: Majid ht/94 [braided, intro]; Dragovich mp/03 [non-Archimedean, adeles]; Bezdek 10 [discrete geometry]; > s.a. special relativity [hyperbolic].
 Types of metric geometries: see 2D, 3D, 4D, differential geometry; euclidean, lorentzian and riemannian geometry; complex structures.
 Other types of geometries: see affine, combinatorial geometry, Finite Geometry, finsler, Graded, non-commutative, projective, symplectic geometry.

History > s.a. euclidean geometry; history of mathematics.
* Origin: It started in Greece as the study of plane and solid figures, and came to be considered as the science of points and their relations in space; Other than the study of regular figures, until Descartes introduced analytic geometry and the invention of the calculus in 1665–1675, a lot of geometry was experimental.
* Kant: Inspired partly by Newton's success to found physics on geometrical principles, thought of geometry as synthetic but a priori; Wrong, as seen after the development of the many non-euclidean geometries in the XIX century.
* Poincaré: Held the view that geometry is a convention and cannot be tested experimentally.
* Lobachevskii, Bolyai, Gauss: Showed that non-euclidean geometries are possible, but used constant curvature (rigid displacements); The distinction between physical and mathematical geometry begins.
* Unified framework: Two proposals were made, Riemann's theory of manifolds, and Klein's Erlangen Programme.
* Erlangen Programme (Klein 1872): A geometry is characterized by an underlying set and a group of transformations acting on it, that are to be considered as equivalences; It won prompt acceptance, encouraged Lie (Lie groups), Poincaré (algebraic topology), Minkowski, and stimulated the conventionalist view of geometry.
* Manifold theory (Riemann): Makes geometry local and introduces g_ab and R_abcd; Perfected by Christoffel, Schur, Ricci-Curbastro, and used in physics by Einstein (but H Hertz would have, if given the time).
@ General references: Heilbron 98; Berger 00 [Riemannian, XX century]; Mlodinow 02; Henderson & Taimina 04; Gray 10 [XIX century]; Holme 10 [and overview]; Ostermann & Wanner 12 [upper-level undergraduate]; Kragh a1205 [attempts to establish links between non-Euclidean geometry and the physical and astronomical sciences, 1830 to 1910]; De Risi ed-15; Biagioli 16 [non-Euclidean geometry in neo-Kantianism]; Smilga 19, Jenkovszky UJP(19)-a1912 [non-Euclidean].
@ Erlangen Programme: in Reid 70; in Torretti 83; in Stewart ThSc(90)may; Kisil in(12)-a1106 [overview]; Goenner a1510-in [influence on physical theories]; > s.a. symmetries in physics.

Techniques in Geometry > s.a. Coarse Structures; curvature of a connection; spectral geometry.
* Constructive solid geometry: A computer graphics technique used for modeling complex objects; Uses cubes as primitive objects, and manipulates them by scaling, stretching, and the binary operations of difference, union, and intersection.
* Integral geometry: Sometimes taken to mean the study of methods for the reconstruction of functions in a real affine or projective space from data on integrals over lines, planes, spheres or other sets.
@ Computational methods: Anderson & Torre JMP(12)-a1103 [symbolic tools for differential geometry]; > s.a. statistical geometry [including computational geometry].
@ Integral geometry: Santaló 76; Palamodov 04.

Applications > s.a. differential geometry; types of metrics [information geometry].
* In physics: Every revolution in physics has brought to the forefront a new type of geometry; The geometry used to describe spacetime in classical physics is Euclidean geometry, in relativistic gravity Riemannian and Lorentzian differential geometry; To describe Hamilton and Lagrange's classical mechanics and classical field theory we use symplectic geometry and variational calculus on jet bundles; Gauge theory is formulated in terms of the geometry of fibre bundles; In quantum theory one deforms the classical structures into non-commutative ones; > s.a. Urs Schreiber's page.
@ In physics: Mackey in(88); Atiyah JMP(95) [quantum physics]; Durham phy/00 [history]; Meschini PhD(08)-a0804 [and relativity and quantum theory]; Hacyan EJP(09) [geometry as an object of experience]; Atiyah a1009-ln [unsolved problems]; Boya IJGMP(12); Eschrig 11; Lavenda 11 [non-euclidean geometries and relativity]; Cariñena et al 15 [geometry from dynamics]; Kerner a1712 [conceptual, historical, Thales]; > s.a. field theory; mathematical physics.
@ Quantum geometry: issue JMP(95)#11; Meschini et al SHPMP(05)gq/04 [pregeometry]; > s.a. discrete geometry; geometry in quantum gravity.
@ Other physics topics: Hélein a0904-conf [new geometries from soldering forms]; Cattaneo et al ed-11 [higher structures]; > s.a. Area Metric; finsler geometry; optics [optical geometry]; unified models.
@ Other applications: Luminet IAU-a0911 [science and art]; Glaeser 12.
> Spacetime geometry: see general relativity; spacetime and spacetime models.

Philosophy of Geometry > s.a. spacetime.
* Idea: One must distinguish between mathematical geometry and physical geometry; The first one is analytic and a priori, the second one synthetic and a posteriori.
* Conventionalist view: Physical geometry depends on conventions; We can assign any geometry to physical spacetime, as long as we choose our rules for measuring lengths and our physical laws accordingly (Poincaré); A criterion for the choice is the disappearance of universal forces (Reichenbach).
* End of XIX century: Trilemma between apriorism, empiricism, conventionalism.
* Helmholtz: Geometry is the study of congruences of rigid bodies; This supported geometric conventionalism.
@ General references: Sexl GRG(70) [conventionalism]; Grünbaum 73, in(77); in Torretti 78; Magnani 01.
@ In relation to special relativity: Schlick 20; Reichenbach 57; Carnap 66.
@ Related topics: Earman GRG(70) [empiricist view]; Carrier PhSc(90)sep [physical geometry].

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