Differentialgeometrie

Those Differentialgeometrie places as Subsection of mathematics the synthesis of and .

Table of contents 1 Subsections 1.1 Classical differentialgeometrie 1.2 Modern differentialgeometrie 1.3 Riemann geometry 1.4 Differential topology 1.5 Theory of the Liegruppen 2 Areas of application 3 Methods 3.1 Koordinatentransformationen 3.2 Kovariante derivative 4 Literature 4.1 Elementary differentialgeometrie 4.2 Abstract variousnesses, Riemann geometry 4.3 General relativity theory 5 Web on the left of

Subsections

Classical differentialgeometrie

The elementary differentialgeometrie is also occupied Curves and in the three-dimensional opinion area and their curvature characteristics. To the classical study objects for example those belong Minimum surfaces, those in nature as forms of Soap skins develop.

Modern differentialgeometrie

The abstract differentialgeometrie develops from that intrinsischen Description of geometrical objects, D.h. the description without resort to a surrounding area. The central term is that the differentiable : a <math>n</math> dimensional diversity is a geometrical object (more exact: to the study of topological characteristics of the regarded variousnesses.

Theory of the Liegruppen

How up , are variousnesses the basis that are based Liegruppen. Liegruppen appear in many places of mathematics and physics as groups of continuous symmetries, for example as turns of the area. The study of the transformation behavior of functions under symmetries leads to Representation theory the Liegruppen.

Areas of application

Application finds the classical differentialgeometrie in that General relativity theory. It makes the forecast possible of phenomena, by that to be confirmed ().

Koordinatentransformationen correspond in that changes of reference systems, from which a phenomenon is observed. This corresponds to different aspects on an event.

The classical differentialgeometrie became already in former times in that and Cartography used. Example is here among other things those from that the terms geodetic line and distances in the area, then one uses cartesian coordinates.

A simpler example is the transition of cartesian coordinates in the level to polar coordinates, with which one can describe a circle line more simply.

<math>f(r, \Phi)=(r\cos \Phi, r\sin \Phi) = (x, y)</math>

The coordinates (x,y) compute themselves out <math>(r, \Phi)</math> as follows:

• <math>x(r, \Phi) = r\cos \Phi</math>
• <math>y(r, \Phi) = r\sin \Phi</math>

x and y become also as component functions of f designated. For this (total) the differentials can be indicated:

<math>\mathrm{d}x=\frac{\partial x}{\partial r}\mathrm{d}r +\frac{\partial x}{\partial \phi}\mathrm{d}\phi = \cos \phi \mathrm{d}r - r\sin \phi \mathrm{d} \phi</math>

<math>\mathrm{d}y=\frac{\partial y}{\partial r}\mathrm{d}r +\frac{\partial y}{\partial \phi}\mathrm{d}\phi = \sin \phi \mathrm{d}r + r\cos \phi \mathrm{d} \phi</math>

One designates Dx,dy,dr, <math>\mathrm{d}\phi</math> as Coordinate differentials. With this example the meaning of D coincides as a differential operator with the meaning of an infinitesimal distance.

Spherical coordinates become also as curvilinear coordinates designated, there it the spacer computation on a curved surface, which kugeloberflaeche, make possible.

A substantial aid of the classical differentialgeometrie are koordinatentransformationen between arbitrary coordinates, in order to be able to describe geometrical structures. Often curvilinear coordinates are used.

From the analysis admitted Differential operators to curvilinear differential operators are extended.

Kovariante derivative

A curvilinear differential operator is z.B. those kovariante derivative, in Riemann area one uses.

Curvilinear differential operators make the definition spaces curved by connecting lines in, for z possible.B. the definition of Geodaeten in the Riemann area. Geodetic lines are the shortest connections between two points on a kugeloberflaeche. The length circles on a ball are examples of geodetic Liníen, not however the parallels of latitude (exception: Equator).

With the help of general koordinatentransformationen those become in the Riemann area <math>\Gamma^\mu_{\alpha\beta}</math> defined.

The Christoffelsymbole goes to that into the definition kovarianten derivative one Vector field .

The kovariante derivative is a verallgemeinerung that partial derivative the flat (Euclidean) area for curved spaces. It reduces in to the partial derivative. In the curved space the kovarianten derivatives of a vector field are exchangeable generally not with one another, their Nichtvertauschbarkeit become the definition of the Riemann curvature tensor used.

A further important term in connection with curved spaces is those Translation. The translation of a vector along a closed curve leads in the curved space to the fact that the shifted vector with its output vector does not cover itself.

Literature

Elementary differentialgeometrie

• Werner Blascke, Kurt light white, Elementary differentialgeometrie, Springer publishing house, 1973

This book is a very beautiful introduction to this subsection of mathematics. It creates the spagat between the geometrical ideas and abstraction.

• Manfred P. DO Carmo: Differentialgeometrie of curves and surfaces. Vieweg & son, 1983

The elementary differentialgeometrie in such a way specified describes. Contains a section over translation.

• E. Welfare: Differential forms. BI science publishing house, 1974

An introduction to the analysis and like it by differential forms to be described can. The book is helpful for the understanding of differentiable variousnesses. It is easily understandably written (relative).

Abstract variousnesses, Riemann geometry

• Rolf walter: Differentialgeometrie. BI science publishing house, 1989

Differentialgeometrie from the point of view of modern mathematics. With a chapter concerning Riemann geometry.

• Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric spaces. Providence, RI, 2001. ISBN 0-8218-2848-7

Standard reference, in particular also for the classification of the halfsimple Liegruppen.

• S. Kobayashi and K. Nomizu, Foundations OF differential Geometry. I, New York 1963

Abstract standard work.

General relativity theory

• Misner, Kip Thorne, John Wheeler: Gravitation

The famous, thousand-lateral "three-man book"; affectionate didactical dressing: each concept formation is descriptive justified, each calculation is carefully motivated.

• H. Stephani: General relativity theory: an introduction to the theory of the gravitational field. Dt. Verl. D. Wiss. Berlin 1977

With chapters over Riemann geometry, tensor algebra and Kovariante derivative, curvature tensor and differential operators.

• Robert M. Forest: General Relativity

General relativity theory mathematical beginning: mathematically cleanly contrary to most other introductions: thus more correctly than intuitive.

• Stephen W. Hawking, G. F. R. Ellis:The Large Scale Structure OF space time

Very mathematical book, singularity theorem (Hawking, Roger_Penrose); concerns itself with the cosmological questions of general relativity theory.

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• Differential geometry homepage