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} T^{n},
rotations and translations) as symmetry group
*G*, and has a globally defined positive-definite bilinear form on T*X* × T*X*,
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 T*X* 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].

@ __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 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. 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.

* __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 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 11 [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].

@ __Erlangen Programme__: in Reid 70; in Torretti 83; in Stewart ThSc(90)may; Kisil 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__: 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; 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]; > 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].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 16
mar
2017