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];
Johnson 18 [geometries and transformations].

@ __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.

* __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].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 2 aug 2018