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 = Rⁿ 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 99 [II]; Benz 05 [classical geometries]; Reid & Szendröi 05 [and topology, II]; Peterson a0708 [directions of research].
@ Erlangen Programme: in Reid 70; in Torretti 83; in Stewart ThSc(90).
@ Diophantine: Lang 60; Hindry & Silverman 00 [intro].
@ Other types: Majid ht/94 [braided, intro]; Dragovich mp/03 [non-Archimedean, adeles].
> Types of metric geometries: see differential geometry, euclidean [including hyperbolic], lorentzian, riemannian geometry.
> Other types of geometries: see affine, discrete geometry, Finite Geometry, finsler, non-commutative, projective, symplectic geometry.

History [> s.a. history of mathematics.]
* Origin: 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 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 cy.
* 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.
* Unification: 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 Herzt would have, if given the time).
@ References: Heilbron 98; Berger 00 [Riemannian, XX cy]; Mlodinow 01; Henderson & Taimina 04.

Techniques in Geometry > s.a. Coarse Structures; spectral geometry; statistical 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.
@ Integral geometry: Santaló 76; Palamodov 04.

Geometry and Physics > s.a. differential geometry; finsler; models of spacetime; optics [optical geometry]; types of metrics [info geometry].
@ General references: Mackey in(88); Atiyah JMP(95) [quantum physics]; Durham phy/00 [history]; Meschini a0804-PhD [and relativity and quantum theory]; > s.a. mathematical physics.
@ Quantum geometry: issue JMP(95)#11; Meschini et al SHPMP(05)gq/04 [pregeometry]; > s.a. geometry in quantum gravity.

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 cy: Trilemma between apriorism, empiricism, conventionalism.
* Helmholtz: Geom 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; Reichenbach 57; Carnap.
@ Related topics: Earman GRG(70) [empiricist view]; Carrier PhSc(90) [physical geometry].

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