Differential Geometry

In Mathematical, the differential geometry is the application of the tools of the differential Calculus to the study of the Géométrie. The objects of basic study are the differential varieties, units having a sufficient regularity to consider the concept of Dérivation, and the functions defined on these varieties.

The differential geometry finds its principal application physical in the Theory of relativity where it allows a modeling of a Courbure of the Espace-temps. One can also quote other applications in traditional physics. In mechanics of the continuous mediums for example, it is useful for the description of the deformations of the elastic bodies, in particular of the beams or the hulls.

Intrinsic and extrinsic point of view

Until the middle of the 19th century, the differential geometry had primarily an extrinsic point of view of about the varieties met, which means that those were defined like a subset of a vector Space (generally $\backslash \; R^n\; \backslash ,\; \backslash !$). For example, one studied the properties of a curve in the plan, or of a surface in the space of dimension three (traditional differential Géométrie).

Work of Bernhard Riemann introduced an intrinsic vision varieties, unceasingly developed since; they are then regarded a “rough” object, and not as part of another. There is no more direction to want “to leave” the variety since it exists independently of any concept of ambient space, and yet one will be able to give a direction to the concepts of tangency , curve , etc

The intrinsic point of view with the advantage of being much more flexible than the extrinsic point of view, would be this only because it does not force to find a space being able “to contain” the variety considered, which can sometimes appear difficult. For example the Bouteille of Klein is a surface (i.e. a variety of dimension 2) but to plunge it in an ambient space it is necessary to choose $\backslash \; R^4\; \backslash ,\; \backslash !$. In the same way, it is not obvious to find a space “containing” the curved space time. However, gained flexibility is translated into an abstraction and a difficulty increased to define the geometrical as the Courbure or topological concepts like the connexity.

Mathematical explanation

The differential geometry covers the analysis and the study of various concepts:

• the study of the varieties
• the tangent fibers and cotangents
• the differential forms
• the derivative external
• Integral S of the p-forms on p-varieties
• the Theorem of Stokes
• derived from Dregs
• the Curve

All are in connection with the analyzes with several variables, but for the geometrical applications, it is necessary to reason without privileging a frame of reference. These concepts distinct from the differential geometry can be regarded as those which include the geometrical nature of the derived second , i.e. the characteristics of the curve.

A differential variety in a topological Espace is a collection of Homéomorphisme S of open units towards a unit sphere R ⁿ such as the open units cover space and that if F , G is homeomorphisms then the function F ^-1 O G of an open subset of the unit sphere towards the unit open sphere is infinitely differentiable. It is said that the function of a variety towards R' is infinitely differentiable if the composition of each homeomorphism results in an infinitely differentiable function starting from the unit sphere open to R . In each point of the variety is a tangent space at this point made up all speeds (direction and intensity) possible and with which it is possible to deviate from this point. For a variety with N dimensions, tangent space in any point is a vector space with N dimensions or, in other words, a copy of R ⁿ. Tangent space has several definitions. A possible definition is the vector space of the ways which pass in this point, quotienté by the relation of equivalence which identifies two ways having the same “Flight Path Vector” in this point (i.e. the same derivative if one composes them with an unspecified chart).

A Champ of vectors is a function of a variety towards the disjoined union of its tangent spaces (the union in itself is a variety known as the tangent Fibré) in such a way that, in each point, the value obtained is an element of tangent space in this point. Such a relation is called section of a fiber. A vector field is differentiable so for each differentiable function, the application of the field in each point produces a differentiable function. The vector fields can be perceived like differential equations independent of time. A differentiable function of real towards the variety is a curve on the variety. That defines a function of real towards tangent spaces: the speed of the curve on each point which constitutes it. A curve is a solution of the vector field if, for each point, the speed of the curve is equal to the vector field in this point.

A linear K-form alternate is an element of the ke power of an antisymmetric tensor of the dual E^* of a vector Space E. a differential K-form of a variety is a choice, in each point of the variety, of such an alternate K-form where E is tangent space in this point. It will be differentiable if the result, after an operation on differentiable K-fields vectorial, is a differentiable function of the variety towards realities.

Branches of the differential geometry

Geometry riemannienne

The Géométrie riemannienne is the study of the metric riemaniennes: such metric is a family of Euclidean products on a differential variety. This additional structure reveals the variety like a Euclidean Espace according to an infinitesimal point of view. It makes it possible to generalize the concepts length of the curve and of measurement of Lebesgue, the analysis of the Gradient of a function, divergence, etc Its strong development during second half of the 20th century is explained by the interest which as well the Géomètre S carried to him as the Analyste S or the Physicien S. Moreover, metric the riemanniennes can be arbitrarily introduced to conclude calculations on the varieties.

Geometry of Finsler

The Géométrie of Finsler is an extension of the geometry riemannienne, which takes all its direction in infinite dimension (for example for the study of the groups of diffeomorphisms on a variety). The main object of study is the Variété of Finsler, id is a differential variety provided with a Métrique of Finsler, a standard of Banach definite on each tangent space.

Symplectic geometry

The symplectic Géométrie is the study of the symplectic forms, i.e of the “not degenerated closed differential forms”. These objects were introduced into the optics of a mathematical formulation of the traditional Mécanique. If the physical motivations go back to Lagrange and Hamilton, the symplectic geometry was formed like field of studies to whole share since the years 1960 and became an active field of research today. Contrary to the geometry riemannienne, of the questions of existence of the structures are posed. The main motors of research are the Conjecture of Arnold, the Conjecture of Weinstein and the Quantification.

Geometry of contact

The Géométrie of contact is the sister of the symplectic geometry in odd dimension. It is primarily about the study of the forms of contact, i.e. differential 1-forms $\backslash \; alpha$ such as $\backslash \; alpha\; \backslash \; wedge\; (D\; \backslash \; alpha)\; ^n$ is a form volume (does not cancel itself in any point). Even if a priori the object of study seems different, “sister” is a doubly justified denomination. On the one hand because the geometries symplectic and of contact present “elementary” results similar. In addition, bus of the Hypersurface S presenting of the structures of contact are omnipresent in symplectic geometry.

Bonds

Internal bonds

• Fibré vectorial

• Géométrie riemannienne
• nonEuclidean Géométrie
• Géométrie of Finsler
• symplectic Géométrie
• Géométrie complexes
• Géométrie of contact
• Groupe of Dregs
• General relativity
• Topologie
• differential Topologie

Wikiversity

• differential Geometry

External bonds

• Course on the curves and surfaces
• Gallery of images
• Notes on the differential geometry

References

Virtual library

• Alain Chenciner; 5 cours on the differential geometry , school of summer “Maths and Brain” of the Institute of Mathematics of Jussieu (2005); pdf.

Bases of the differential geometry

• Differential Geometry off Curves and Surfaces of Manfredo C Carmo (1976).
• Riemannian Geometry of Manfredo Perdigao C Carmo, Francis Flaherty (1994)
• Geometry from has Differentiable Viewpoint of John McCleary (1994)
• has off First Course in Geometric Topology and Differential Geometry of Ethan D. Bloch (1996)
• Modern Differential Geometry Curves and Surfaces with Mathematica , 2nd ED. of Alfred Gray (1998)

Random links:

Mang (symbol) | Second battle of El Alamein | End Mac Cumaill | Church of Christ in Congo | National String Instrument Corporation | Bennebroek

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org