Real manifold

In mathematics, real submanifolds are a concept from calculus and differential geometry. Since real manifolds are subsets of a Euclidean space, they inherit from this many features such as the ability to measure distances. However, one can real submanifold as abstract differentiable manifolds (without surrounding space ) look. The equivalence of the two viewpoints is ensured by the embedding theorem of Whitney.

Submanifolds of Euclidean space

Selected examples where the submanifolds play a role are:

• Constrained optimization
• Mechanical systems with constraints
• Differential-algebraic systems of equations in the numerical network analysis in electrical engineering

In all these applications, the amount of the observed points is limited from the outset to a subset of that by diffeomorphisms on areas can be mapped with a local. This subset is called -dimensional submanifold of. With the help of diffeomorphisms one can expect the same on the submanifold in the differential geometric sense as in areas of.

Usually the amount is described by constraints. This means that just contains those points that set with a continuously differentiable function with the equation

. meet In addition, even required, that is a regular value of, ie, the Jacobian matrix of all points has the maximum rank.

The last condition ensures the applicability of the implicit function theorem. This means that there will be any point is an environment of where the points are already clearly parameterized by coordinates. The picture that is projected onto the coordinates required for the parameterization, is an example of a card images and the associated map area. Since, for every point a map image, you can completely overlap with the associated maps areas. A lot of these cards, with their maps areas you can cover, is an example of an atlas.

With the help of the card images can be locally as expected. The motivation to the natural number of dimension is called and is referred to as the manifold -dimensional.

Example

The unit sphere in is described by the continuously differentiable function by the equation. The Jacobian matrix has to with its maximum rank one. So is

An (n - 1) - dimensional submanifold of. At each point of at least one coordinate is not equal to zero. For you can with the amount

Use as a map area and with the amount

The illustrations

With

Then can be used as maps for these areas.

The easiest way to illustrate this procedure for the one-dimensional unit sphere in. The picture to the four map areas are drawn as thick solid lines. The union of the map areas covering the entire unit sphere, ie, these cards together an atlas. The respective areas corresponding to the maps flattener are indicated by a small arrow. The images of the map areas are dashed thick.

For the two-dimensional unit sphere in one already needs two coordinates for unambiguous parameterization of the points on the maps areas. For example, one chooses for the amount and as a map image.

Also the Mobius band has properties as a local area of ​​the can and should therefore also be referred to as a two-dimensional differentiable submanifold. If the Möbius strip represented as archetype of a regular value of a continuously differentiable function, then the perpendicular steady on standing gradient of this function should everywhere pointing in one direction ( eg as a point away from the front ). However, this is not because the Möbius strip has no front or back. Why must the definition of differentiable submanifold of be taken more generally.

General definition of a submanifold of Rn

A set is a -dimensional - times continuously differentiable submanifold of if there is any point to an environment and a - times continuously differentiable function with regular value 0, such that.

Important statements

Equivalent of this is: A set is then exactly one - times continuously differentiable submanifold of if there is any point to a local flat makers to that, there are a environment and a diffeomorphism with for all.

A regular parametric representation is a continuously differentiable function that an area of the maps and their Jacobi matrix for each parameter has the maximum rank.

Is a local flat makers a manifold, then is a regular parametric representation, the at least the portion of parameterized. In this case, with projected on the principal components of local flat -maker.

Local can be obtained by regular parametric equations define manifolds: is it a regular parametric representation and arbitrary, then there exists a neighborhood, so that the image of differentiable under a submanifold represents the.

Example

The right illustrated with immersion is an example that the above statement is not necessarily on the full image of an immersion can be generalized ( even not if, as in this example, the immersion is injective ). The amount is local to the point not diffeomorphic to an interval of the real axis and thus does not constitute a one-dimensional submanifold of dar.

Tangent / tangent / tangent

Let be a -dimensional differentiable submanifold of and. A vector is called tangent to the point, if it is a differentiable curve and there.

Is considered as a trajectory of a moving particle on the bottom manifold so that particles passed the time the point of interest with the right speed.

The set of all the tangent vectors at a point -dimensional linear space and is referred to as the tangent to the point.

By definition, leaves the submanifold in a neighborhood of the point represented as a regular zero of a function. Be an arbitrary continuously differentiable curve. Since this runs on the manifold, it satisfies the equation. Results derived according to the point, from which follows:

The tangent arises just as the core to associated Jacobian matrix, that is, it is true.

Did they give a (local) regular parametric representation in the maps, so can a parameter point to represent the tangent space in as a full image of the associated Jacobian matrix:

The relation that maps each point all tangent vectors at this point, is called the tangent bundle of.

Be at least twice continuously differentiable submanifold of and arbitrary. From a local representation of in a neighborhood of a local representation of can be constructed:

This is one -dimensional (at least once ) continuously differentiable submanifold of ( in the sense of the usual identification of the final).