Differential geometry
The differential geometry provides as a branch of mathematics, the synthesis of analysis and geometry dar.
- 3.1 Coordinate Transformations
- 3.2 Covariant derivative
- 3.3 curvature tensor
- 4.1 Elementary Differential Geometry
- 4.2 Abstract manifolds, Riemannian geometry
- 4.3 Differential Geometry of Defects
Historical development and current areas of application
A number of fundamental contributions to differential geometry derived from Carl Friedrich Gauss. During this time the math was still strongly associated with various application areas. Important results were obtained with this theory while in the fields of cartography, navigation and geodesy. There developed among others the map projection theory, from which the terms and Gaussian curvature geodesic come. In addition, C. F. presented Gauss already the question of whether measured by bearing angle sum of a very large triangle actually is exactly 180 degrees, and thus proves to be a pioneer of modern differential geometry.
The modern differential geometry finds its application mainly in the general theory of relativity and in satellite navigation. It allows the description of phenomena such as astronomical light deflection or perihelion of Mercury, which can be confirmed by experiments. Coordinate transformations correspond in relativity theory the change of reference systems, out of which a phenomenon is observed. This therefore corresponds to different states of motion of the measuring apparatus and the observer.
Another important field of application is in the theory of defects and plasticity.
Subregions
Elementary Differential Geometry
The initial work on differential geometry deal with both curves and two-dimensional curved surfaces in three-dimensional real space of intuition. Historically, it was first possible with Gauss's work to capture the curvature, for example, the two-dimensional surface of a sphere and quantitatively.
Another motivation for the development of the elementary differential geometry came here also from the mathematical problem of minimal surfaces. The naturally occurring soap films can be described as minimal surfaces. The shape or mathematical representation of these surfaces can thereby develop the methods of the calculus of variations. The geometric characteristics of these surfaces, such as curvature or spacing between any two points on a minimum area, however, are rather calculated using the methods of differential geometry.
Differential topology
The differential topology is the basis for most modern branches of differential geometry. In contrast to the basic differential geometry the geometrical objects are intrinsically in the described differential topology, that is the definition of the properties is made without recourse to a surrounding space. The central concept is that of differentiable manifold: One -dimensional manifold is a geometric object (more precisely, a topological space ) that looks locally like - dimensional real space. The classic example that motivates the terminology, is the earth's surface. In small cut-outs they can be described by maps, ie small parts " look like " the plane. However, the entire surface of the earth can not identify with the plane. In addition, differentiable manifolds carry a structure that makes it possible to speak of differentiable functions. Differentiable this structure makes it possible to apply to the card locally analytical methods. In addition, one can investigate the diversity globally as a topological space. So tried the differential topology connections between the local analytical and establish the global topological properties. An example of such a link is the set of de Rham.
Riemannian geometry
On a differentiable manifold, there is no predefined length measurement. If it is given as an additional structure, it is called Riemannian manifolds. These manifolds are the subject of Riemannian geometry, which also examines the associated notions of curvature, the covariant derivative and parallel transport on these quantities.
However, a generalized metric structure ( with possibly negative intervals ) were examined, these manifolds are Lorentz, semi- or pseudo- Riemannian manifolds also called. A special case are the solutions of Einstein's field equations, these hot Einstein manifolds ..
Semi- Riemannian differential geometry
If one replaces the condition of positive definiteness by the weaker condition of non Entartet Slope for the metric of a Riemannian manifold, we obtain a semi- Riemannian manifold. A special case are the Lorentz between manifolds of general relativity.
Finsler geometry
Subject of the Finsler geometry Finsler manifolds are, that is manifolds whose tangent space is equipped with a Banachnorm, ie a mapping with the following properties:
Finsler manifolds also play in theoretical physics as general candidates for the structural description of space-time a role.
Symplectic Geometry
Instead of a symmetric nondegenerate bilinear form an anti-symmetric non-degenerate bilinear form ω is given. If this is also still closed, ie d Ⓜ = 0, is called a symplectic manifold. Because a symplectic vector space has dimension necessarily straight, even symplectic manifolds have just dimension. The first important finding is the set of Darboux symplectic manifolds according to the locally isomorphic to T * Rn are. Thus, there is in contrast to semi- Riemannian manifolds no ( non-trivial ) local symplectic invariants (except the dimension), but only global symplectic invariants. As a generalization include the Poisson manifolds that do not have bilinear form, but only an antisymmetric bivector. This induces a Lie bracket between functions. Symplectic geometry has applications in Hamiltonian mechanics, a branch of theoretical mechanics.
Contact geometry
The analogue of the symplectic geometry of manifolds is ungeraddimensionale contact geometry. A contact structure on a 2n 1- dimensional manifold M is a family H of hyperplanes of the tangent bundle that are maximally non- integrable. Local can be represented as a core of an α 1-form these hyperplanes, ie
Conversely, a contact form is locally uniquely determined by the family H, up to a nonzero factor. The Nichtintegrabilität means that d.alpha restricted to the hyperplane is non- degenerate. If the family H can be described globally by a 1- form α, then contact form α iff
It is a theorem analogous to the Darboux theorem for symplectic manifolds, namely, that all contact manifolds of dimension 2n 1 are locally isomorphic. This is also available in the contact geometry only global invariants.
Complex Geometry and Kählergeometrie
Complex geometry is the study of complex manifolds, ie manifolds that look locally like Cn and whose transition functions are complex - differentiable ( holomorphic ). Because of the analytical properties of complex - differentiable functions, one has here frequently uniqueness properties of the continuation of local functions / vector fields. That is why you usually rely on global studies on the theory of sheaves. An almost - complex structure on a smooth manifold is a map J: TM → TM such that J2 = -1. This means that all almost - complex manifolds of even dimension. The difference between a near- complex and a complex manifold is the integrability of the almost - complex structure. This is measured by the Nijenhuis tensor NJ.
A Hermitian manifold is a complex manifold with a Hermitian metric g on the tangent bundle of complexified real. G in particular, must be compatible with the complex structure of J, in particular
To be particularly rich in texture to Hermitian manifolds have proven their hermitian metric are also compatible with a symplectic form, ie
In this case one speaks of a Kählermannigfaltigkeit.
Finally, the Cauchy -Riemann Geometry is concerned with bounded complex manifolds.
Theory of Lie groups
Just as groups are based on quantities manifolds are the basis of Lie groups. Named after Sophus Lie Lie groups occur in many areas of mathematics and physics as a continuous symmetry groups, for example, as groups of rotations of the space. The study of the transformation behavior of functions under symmetries leads to the representation theory of Lie groups.
Global Analysis
Global Analysis is also a branch of differential geometry that is closely related to the topology. Sometimes it is called the sub-region and Analysis on Manifolds. In this mathematical research area of ordinary and partial differential equations on differentiable manifolds are investigated. How to find in this theory local methods of functional analysis, the micro- local analysis and the theory of partial differential equations and global methods from the geometry and topology application. Since this mathematical branch used compared to the other fields of differential geometry very many methods of analysis, it is partially understood as a branch of analysis.
Already the first work on differential equations contained aspects of global analysis. Thus, the studies of George David Birkhoff in the field of dynamical systems and the theory of geodesics of Harold Calvin Marston Morse are early examples of methods of global analysis. Central results of this mathematical part of the area are the works of Michael Francis Atiyah, Isadore M. Singer and Raoul Bott. Particularly noteworthy here are the Atiyah-Singer index theorem and the Atiyah - Bott fixed point theorem, which is a generalization of Lefschetz'schen fixed point theorem from topology.
Methods
Coordinate transformations
Coordinate transformations are an important tool of differential geometry to enable the adaptation of a problem to geometric objects. For example, to distances are examined on a spherical surface, as generally spherical coordinates may be used. Considering Euclidean distances in space, you use the other hand rather Cartesian coordinates. Mathematically, noted that coordinate transformations are always bijective, any number of times continuously differentiable mappings. Thus there is always the inverse of the observed coordinate transformation.
A simple example is the transition from Cartesian coordinates to polar coordinates in the plane. Each position vector of the two-dimensional Euclidean space can be in this view by the coordinates and expressed in the following manner
X and y are referred to as components of f functions. They are calculated according to a function of the two coordinates:
Are now quite generally all the coordinates of the new coordinate system is kept constant to a coordinate and changing the single coordinate of the definition domain, occur in Euclidean space lines, also referred to as the coordinate lines. In the case of the polar coordinate as shown occur at a constant coordinate of concentric circles with a radius about the origin of the Euclidian coordinate system. At constant coordinate arise half-lines that start at the origin of the Euclidean coordinate system and run after. Using these coordinate lines can be in an obvious manner for each point of Euclidean space a new, spatially rotated and re -angled coordinate system defined. One speaks therefore in polar coordinates from rectangular coordinates. The axis of the rotated coordinate system are straight lines, the coordinates of the tangents passing through the point. The basis vectors of this space-dependent and rectangular coordinate systems can be directly via the partial derivatives of the position vector, calculated in accordance with the above representation, according to the variable coordinates. About the partial derivatives can also be the total differentials of the position vector specify:
The differentials dx, dy, dr, can be referred to as the coordinate differentials. In this example, associated with the differential operator "d" infinitesimals do not always have the significance of a distance. It rather shows relatively easy, that applies to the distances in the radial or azimuthal direction that is indeed, but; ie only the prefactor " " is obtained by integrating over from 0 to a known quantity of the dimension 'length', namely the circumference.
The polar coordinates or three-dimensional generalization of the spherical coordinates are also referred to as curvilinear, as it on a curved surface, such as the spherical surface to enable distance calculation. It is - as with other standard examples, such as the cylindrical coordinates, the elliptic coordinates, etc. - To curvilinear orthogonal coordinates (see also: Curvilinear coordinates).
An essential tool of classical differential geometry are coordinate transformations between any coordinates to describe geometric structures.
The known from calculus, formed with the size differential operators can be relatively easily extended to curvilinear orthogonal differential operators. For example, apply in general orthogonal curvilinear coordinates when using three parameters and corresponding unit vectors in the direction of the following relationships with sizes that are not necessarily constant, but of, and may depend on:
The points indicated by two additional terms arising from the first term by cyclic permutation of the indices. denotes the Laplace operator. It can be composed of the scalar -valued div operator and the vector -valued level operator in accordance with
In which
The formula for the divergence of the coordinates based on independent representation
Being integrated over the closed bounding surface. denotes the corresponding outer normal vector, the corresponding infinitesimal surface element. In the most general case - that is, for non-orthogonal curvilinear coordinates - you can use this formula also.
Covariant derivative
Overall, based on not necessary orthogonal curvilinear coordinate derivative operators are eg the covariant derivatives, which are used eg in Riemannian spaces where it in a specific way from the " inner product", ie from the so-called " metric fundamental form " of the space, depend. In other cases, however, they are independent of the existence of a local metric or can be specified externally even, for example, in manifolds " with Konnexion ".
Etc. They allow the definition of connecting lines in curved spaces, such as the definition of geodesics in Riemannian space. Geodesic lines are the locally shortest routes between two points in these spaces. The length of circles on a sphere are examples of geodesic lines, but not the width circles (except the equator).
With the help of general coordinate transformations, the Christoffel symbols are defined in the Riemannian space. These are, according to the given below basic definition, explicitly in the calculation of the covariant derivative of a vector field a.
The covariant derivative is a generalization of the partial derivative of the flat ( Euclidean ) space for curved spaces. In contrast to the partial derivative it receives the Tensoreigenschaft; in Euclidean space it is reduced to the partial derivative. In curved space the covariant derivatives of a vector field in general do not commute with each other, their Nichtvertauschbarkeit is used to define the Riemann curvature tensor.
Another important concept in the context of curved spaces is the parallel displacement. The covariant derivative of the components of a vector in a parallel shift is zero. Nevertheless, the parallel displacement of a vector along a closed curve in the curved space can cause the shifted vector does not coincide with its output vector.
The corresponding formalism is based on the requirement that you write vectors as a sum, with may (namely just at previous " parallel transport " ) is not the components, but only the basic elements of change, after the obvious rule:. Covariant and partial derivative, usually written by a semicolon or comma, so different, and that applies:
Of course, in manifolds with additional structure (eg, in Riemannian manifolds, or in the so-called gauge theories ), this structure must be compatible with the transmission. This gives additional relations for the Christoffel symbols. For example, the distance and angular relationships of two vectors may be in Riemannian spaces with parallel shift does not change, and the Christoffel symbols are calculated accordingly in a certain way solely from the metric structure.
Curvature tensor
The above-mentioned space curvature is obtained analogously: If the basis vector in the mathematically positive sense ( counterclockwise ) until an infinitesimal distance in direction and then moves an infinitesimal distance in - direction, we obtain a result which we can write in the form. With change in the order, ie with opposite directions of rotation, you get the opposite result. The difference can thus be of a size, which arises from the Christoffel symbols, write in the following form:
In parallel displacement of the vector results accordingly: The components form the curvature tensor. (In the so-called Yang-Mills theories, this term is generalized. )