Lexicon of the geometry riemannienne

The Géométrie riemannienne is a field of mathematics studying the properties of the riemanniennes varieties. This page briefly points out the definitions of the recurring terms met.

With

Application conforms: Between two riemanniennes varieties, application which preserves the angles; in an equivalent way application which transports metric in metric in conformity;
exponential Application: Differentiable application $TM \ rightarrow M$ defined naturally for any riemannienne variety supplements, characterized by what $exp (TV)$ is the geodetic one of which speed at time 0 is v ;

B

C

Center of mass
osculatory Circle
Field of Jacobi
Field of Killing
Class of Chern
Convexity
bisectionnelle Curve
Curve of Gauss
negative Curve
Curve of Ricci
sectionnelle Curve
Growth of a group
Cut-locus of a point m of a riemannienne variety: Together (negligible) of points N for which there is not unicity of geodetic minimizing;

E

Space of Hadamard: Riemmannienne variety or spaces metric simply related strictly negative curve.
homogeneous Space: Variety on which a group of Dregs acts transitively.
symmetrical Space: Riemannienne variety admitting in any point at least an involution.

F

Flaky preparation riemannien: Flaky preparation of a variety in riemanniennes varieties;
Fiber normal: for a subvariety NR of a variety riemannienne M , fiber vectorial on NR are the fiber in X is the orthogonal one with T _X NR ;
Fiber riemannien: Vectorial Fibé provided with metric a riemannienne;
geodetic Flood: Differentiable flood on tangent space or cotangent of a riemannienne variety, or on fiber in spheres corresponding, defined by the dynamics of the Géodésique S;
Function of Busemann: Function continues definite on a space (riemannienne variety or spaces metric) with limited negative curve intervening in the compactification; the functions of Buseman form the Sphère ad infinitum;
harmonic Form: Form differential whose Laplacian is null;
Form of Kähler:
Formula of the traces of Selberg:

G

Geodetic: Curve locally minimizing the distance on a riemannienne variety;
Geodetic closed: Geodetic periodical;
Euclidean Geometry: geometry of an Euclidean space;
Geometry riemannienne: Geometry of a riemannienne variety;
hyperbolic Group

H

Holonomie
Horosphère

I

Identities of Bianchi: Remarkable identity relating to coubure of the connxion of Levi-Civita;
Inequality of Bishop-Gromov: Estimate on the volume of the balls of a riemannienne variety according to estimates on the curve of Ricci;
isoperimetric Inequality: Any inequality giving an increase of riemannien volume locked up by a hypsersurface according to the volume of the latter;
Involution: Isometry on a riemannienne variety fixing a point and whose differential in this point is - Id;
Isométrie: Between two riemanniennes varieties, differentiable application sending metric riemannienne on metric riemannienne; or in manner are equivalent, continuous application preserving the distances associated;

L

Laplacian: Definite differential operator on any riemannienne variety;

M

Metric of Metric Carnot-Carathéodory
riemannienne: Collection of positive definite forms bilinear symmetrical definite on tangent spaces of a variety, with a certain regularity depending on the context;
Brownian Movement: ???
Metric of Einstein:

NR

Number of Betti: Dimensions of spaces of cohomology of Rham;

P

Plunging riemannien: Plunging preserving metric the riemannienne.
Problem of Dirichlet

Q

Quasi-isométrie : Applications (not necessarily continuous) between riemanniennes varieties or metric spaces which do not dilate the distances excessively.

R

Ray of convexity
Ray of injectivity: larger ray such as the exponential application is injective on the corresponding tangent balls;
Coating riemannien: Universal coating of a riemannienne variety provided with metric drawn behind;
Rigidity of Mostow

S

Spectrum of the Laplacian
Spinor
Symbol of Christoffel: Symbols allowing to express in local charts the connection of Levi-Civita;
Systole (mathematics)

T

Theorem of Abresch-Meyer
Theorem of Bishop
Theorem of Bonnet-Schoenberg-Myers
Theorem of Brown-Minkowski
Theorem of comparison
Theorem of Gauss-Bonnet
Theorem of Hadamard-Cartan: theorem affirming that the universal coating of a riemannienne variety supplements negative curve is difféomorphe with a ball;
Theorem of Hopf-Rinow
Theorem KAM
Theorem of Myer: estimate on the diameter of a riemannienne variety supplements in positive curve;
parallel Transport

V

hyperbolic Variety
Variety kählérienne
Lorentzian Variety
Variety riemannienne
riemannien Volume: form volume defined on any riemannannienne variety directed being worth 1 on any directed orthonormée tangent basis; or measures positive associated;

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