Geometry riemannienne
The geometry riemannienne is a branch of mathematics named in the honor of the mathematician Bernhard Riemann who introduced the concept founder of variety. It extends the methods of the analytical Geometry by using local Coordonnées to carry out the study of curved spaces on which exist concepts of angle and length.
The most notable concepts of the geometry riemannienne are the Courbure studied space and the Géodésique S, curves solving a problem of shorter way on this space. More generally, the purpose of the geometry riemannienne is the local study and total of the varieties riemanniennes, i.e. the differential varieties provided with a Métrique riemannienne, even of the vectorial fibers riemanniens.
There exist also varieties pseudo-riemanniennes, generalizing the riemanniennes varieties, which remain rather close about it by many aspects, and which in particular make it possible to model the Espace-temps in physics.
History
The first step of the geometry riemannienne goes back incontestably to work of Bernhard Riemann with the nineteenth century.
The geometry riemannienne strongly developed during second half of the 20th century. But the first work in this field merges with the birth of the concept of differential variety.
The framework
August 1stProblems of shorter ways
Geodetic: aspects room and total
August 1stParallel transport
August 1stCurve and topology
Positive curve
August 1stNegative curve
August 1stReferences
-
- As indicates it its title, the large French geometrician invites us here to a long (824 pages) panoramic walk in the world of the Riemannienne geometry. The various results for the majority are given without detailed demonstrations, but with the suitable references for the reader who would wish to put “the hands in dirty oil”. The final chapter gives the technical bases of the field.