Density on a manifold

A density bundle is a special case of a vector bundle and is being studied in the mathematical branch of differential geometry. With this bundle you can generalize some known from analysis objects on manifolds. So one can define similar as with differential forms a coordinate- invariant integral term on manifolds. We find with the help of this bundle generalizations of Lp- spaces and the distribution spaces on manifolds.

• 4.1 Definition
• 4.2 Properties
• 4.3 L1 - space
• 4.4 Lp- spaces

• 5.1 density bundle over the real space
• 5.2 Riemannian density
• 5.3 tensor
• 5.4 distributions

Definition

R density of

Be a real, - dimensional vector space and is listed with the n-th exterior power of the vector space. For each to define a R- density as a function, so that

Valid for all and for all. The vector space of densities is quoted at.

R- density bundle

Be a smooth -dimensional manifold and a real number. With the space of global sections is recorded on a vector bundle.

Analogous to the above definition is a density on a manifold a picture

With

For all and for all smooth functions.

The vector bundle of densities is then defined by

With the tangent bundle is called.

Pullback

For induces a smooth map between two smooth - dimensional manifolds a pullback

Which for all by

Is defined. This is the push forward of, and are real submanifolds then the Jacobian of.

Dual space

Integration on manifolds

Therefore one densities are especially important because they can be integrated on manifolds (coordinate -independent). Their advantage over differential forms, which also have this property is that you can integrate densities on non- orientable manifolds.

Definition

So be a smooth manifold and is a 1- density. Then, the integral over is defined as follows. Let be a finite family of maps that cover. And be a subordinate partition of unity. Then set

The right side is independent of the choice of the map and selection of a partition of unity.

Properties

• The integral is invariant under diffeomorphisms. That means for every smooth manifold and the same dimension and each diffeomorphism and each 1- density applies
• The integral is local, that is, for each subset and each 1- density applies
• For each The right-hand integral is a normal Lebesgueintegral a smooth function with compact support.

L1 - space

Be a measurable 1- density with compact support. The integral exists, so a cut is called its norm by

Is given. The completion of this quantity with respect to the given norm will provide the space is the manifold is compact, then the completion does nothing.

Lp- spaces

Be now and and one of the two densities have compact support. Then, due to the two property from the dual space section and has compact support. Thus, it is integrable.

Is integrable then one speaks of an analog interface whose standard by

Is given. The completion will provide the space Also back for two property from the dual space portion is to the space with the dual space

Examples

Density bundle over the real space

Be the manifold under consideration. The tangent bundle is a trivial vector bundle, and therefore exist in the density bundle global sections. Be the canonical basis of, then is a basis of. There is then a smooth nowhere vanishing section defined by

Is defined. For any smooth map is a smooth 1- density. The object can be understood as the Lebesgue measure.

Be a smooth diffeomorphism, then applies

The Jacobian matrix denoted by. This relationship is also found in the coordinate transformation of integrals, compared to also transform set.

Riemannian density

Be an n-dimensional Riemannian manifold, then there exists an orthonormal frame with respect to the Riemannian metric on the tangent bundle. The uniquely determined global average with

Ie Riemannian density. This section exists without condition always.

Tensor

Replace in the definition of the tangent bundle by the Tensorbündel Then it is called the induced density bundle the - Tensordichtebündel. In case the elements are called tensor fields.

Distributions

Since you can as described earlier in the article integrate 1- densities on subsets of a manifold, this now allows distributions to be defined on manifolds. Be the space of smooth sections with compact support. So you can a distribution of induced

Defined by

For this reason it is

This is the space of smooth sections with compact support, which is defined analogously to the space of test functions with compact support. The space of distributions is then defined analogously to real analysis as topological dual space. So you set