# Differentialgeometrie

the Differentialgeometrie represents the synthesis of analysis and geometry as subsection of mathematics .

## subsections

### classical Differentialgeometrie

the elementary Differentialgeometrie is occupied with curves and surfaces in the three-dimensional opinion area and their curvature characteristics. To the classical study objects for example the minimum surfaces belong,in nature as forms of soap skins develop.

### modern trend Differentialgeometrie

the abstract Differentialgeometrie develops from the intrinsischen description of geometrical objects, i.e. the description without resort to a surrounding area. The central term is that the differentiable diversity: one [itex] n< /math> - dimensional diversityis a geometrical object (more exact: a topological area), which looks local in approximately like [itex] the n< /math> - dimensional real area. The classical example, which motivates also the terminology, is the earth's surface: In small cutouts it can be described by maps, i.e. small parts “look how”the level. In order to receive however an overall view of the earth, still the map changes must be described: which parts of two maps correspond themselves? The attribute differentiably refers now to the fact that these map changes should be differentiable illustrations. It makes possible, from differentiable functions on the diversityto speak, and the analysis becomes to a certain extent the local theory, whose global correspondence is the Differentialgeometrie.

### Riemann geometry

on a differentiable diversity does not give it a pre-defined linear measurement. If it is given as additional structure, one speaks of riemannschen variousnesses. They are articleriemannschen geometry, also the terms of the curvature resulting from this structure, which derivative and the translation kovarianten examined.

### differential topology

the differential topology uses means of the Differentialgeometrie and the topology for the study of topological characteristics of the regarded variousnesses.

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### Theory of the Liegruppen

as groups on quantities , are variousnesses the basis of the Liegruppen are based. 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 leadsto the representation theory of the Liegruppen.

the classical Differentialgeometrie in general relativity theory finds areas of application application. It makes possible to be confirmed the forecast by phenomena, those by the experiment (light diverson, Periheldrehung Merkur).

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

The classical Differentialgeometrie was used already in former times in geodesy and cartography. Example is among other things the map projection teachings from that the terms geodetic line and Gauss 'sche curvature comes here.

## methods

### Koordinatentransformationen

Koordinatentransformationen are an important tool of the Differentialgeometrie, in order to make the adjustment possible of a problem definition to geometrical objects.

If one wants to examine distances on a Kugeloberfläche, then one spherical coordinates to use, regards one Euclidean distances in the area, then usedone 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.

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

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

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

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

[itex] \ 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>
[itex] \ 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 D x, D y, D r, [itex] \ 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 are called also curvilinear coordinates, there them the spacer computation upa curved surface, which Kugeloberfläche, 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 differential operators admitted on curvilinear differential operators are extended.

### Kovariante derivative

incurvilinear differential operator e.g. is. the kovariante derivative, which is used in the Riemann area.

Curvilinear differential operators make the definition possible spaces curved by connecting lines in, e.g. the definition of geo data in the Riemann area. Geodetic lines are the shortest connections between two points on a Kugeloberfläche. ThoseLength circles on a ball are examples of geodetic Liníen, not however the parallels of latitude (exception: Equator).

With the help of general Koordinatentransformationen the Christoffelsymbole becomes math \ Gamma^ <\> mu_ {\ alpha \ beta} /math in the Riemann< area> defined.

The Christoffelsymbole enters definition of the kovarianten derivative of a vector field .

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

A further more importantlyTerm in connection with curved spaces is the 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

• William Blaschke,Light white, elementary Differentialgeometrie, Springer publishing house, 1973 this
book is briefly 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
describethe elementary Differentialgeometrie in such a way specified. 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 easy (relative)understandably written.

### 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

the famous, thousand-lateral “three-man book”; affectionate didactical dressing: everyoneConcept 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 theorymathematical beginning: mathematically cleanly contrary to most other introductions: thus more correctly than intuitive.
very mathematical book, singularity theorem (Hawking, Roger Penrose); concerns itself with the cosmological questions thatGeneral relativity theory.

## Web on the left of

 Wikibooks: Differentialgeometrie - learning and teaching materials