Density bundle

A density bundle is a special case of a vector bundle and is examined in the mathematical sub-area of differential geometry . With the help of these bundles one can generalize some objects known from analysis to manifolds . Like with differential forms, one can define a coordinate-invariant integral term on manifolds. With the help of these bundles one finds generalizations of the L ^p spaces and the distribution spaces on manifolds.

definition

r density

Is a real, -dimensional vector space and is the nth outer power of the vector space recorded. For each one defines an r-density as a function such that ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle \Lambda ^{n}V}$ ${\displaystyle V}$ ${\displaystyle r\in \mathbb {R} }$ ${\displaystyle f\colon \Lambda ^{n}V\to \mathbb {R} }$

${\displaystyle f(\lambda u)=|\lambda |^{r}f(u)}$

applies to everyone and everyone . The vector space of the densities is noted with . ${\displaystyle u\in \Lambda ^{n}V\backslash \{0\}}$ ${\displaystyle \lambda \neq 0}$ ${\displaystyle r}$ ${\displaystyle |V|^{r}}$

r-density bundle

Let be a smooth , -dimensional manifold and a real number. With , the space of the global sections is noted on a vector bundle . ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle r\in \mathbb {R} }$ ${\displaystyle \Gamma ^{\infty }}$

Analogous to the definition above, a density on a manifold is a mapping ${\displaystyle r}$

${\displaystyle \mu \colon \Gamma ^{\infty }(\Lambda ^{n}(M))\to C^{\infty }(M)}$

With

${\displaystyle \mu (\lambda u)=|\lambda |^{r}\mu (u)}$

for everyone and for all smooth functions . ${\displaystyle u\in \Lambda ^{n}TM\backslash \{0\}}$ ${\displaystyle \lambda \colon M\to \mathbb {R} \backslash \{0\}}$

The vector bundle of densities is then defined by ${\displaystyle r}$

${\displaystyle |\Lambda ^{n}(M)|^{r}:=|\Lambda ^{n}(TM)|^{r}.}$

With that is tangent referred. ${\displaystyle TM}$

Pullback

→ Main article : return transport

For a smooth mapping between two smooth -dimensional manifolds induces a pullback ${\displaystyle r\geq 0}$ ${\displaystyle \phi \colon M\to N}$ ${\displaystyle n}$

${\displaystyle \phi ^{*}\colon \Gamma ^{\infty }(|\Lambda ^{n}(N)|^{r})\to \Gamma ^{\infty }(|\Lambda ^{n}(M)|^{r}),}$

which for everyone through ${\displaystyle e\in C^{\infty }(\Lambda ^{n}TM)}$

${\displaystyle (\phi ^{*}\mu )(e)=\mu (\det(\phi _{*})\cdot e)=|\det(\phi _{*})|^{r}\mu (e)}$

is defined. Where is the push forward of , are and submanifolds so is the Jacobian matrix of . ${\displaystyle \phi _{*}}$ ${\displaystyle \phi }$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \phi _{*}}$ ${\displaystyle \phi }$

Dual space

1. Be a smooth manifold again. Since the vector space of the 0 densities consists only of the constant functions, the following applies for the corresponding density bundle ${\displaystyle M}$ ${\displaystyle |\Lambda ^{n}T_{p}M|^{0}}$

   ${\displaystyle |\Lambda ^{n}(M)|^{0}\cong M\times \mathbb {R} .}$

2. For true isomorphism ${\displaystyle \alpha ,\beta \in \mathbb {R} }$

   ${\displaystyle |\Lambda ^{n}(M)|^{\alpha }\otimes |\Lambda ^{n}(M)|^{\beta }\cong |\Lambda ^{n}(M)|^{\alpha +\beta }.}$

3. From the properties 1. and 2. it follows and therefore is the dual space of and one writes
   ${\displaystyle |\Lambda ^{n}(M)|^{\alpha }\otimes |\Lambda ^{n}(M)|^{-\alpha }\cong M\times \mathbb {R} }$
   ${\displaystyle |\Lambda ^{n}(M)|^{\alpha }}$ ${\displaystyle |\Lambda ^{n}(M)|^{-\alpha }}$
   ${\displaystyle (|\Lambda ^{n}(M)|^{\alpha })'\cong |\Lambda ^{n}(M)|^{-\alpha }.}$

Integration on manifolds

One-densities are particularly important because they can be integrated on manifolds (independent of coordinates). Their advantage over differential forms, which also have this property, is that densities can also be integrated on non- orientable manifolds.

definition

So be a smooth manifold and be a 1-density. Then the integral of over is defined as follows. Be a finite family of cards that cover . And be a subordinate decomposition of the one . Then sit ${\displaystyle M}$ ${\displaystyle \mu \in \Gamma _{c}^{\infty }(|\Lambda ^{n}(M)|^{1})}$ ${\displaystyle \textstyle \int _{M}\mu }$ ${\displaystyle \mu }$ ${\displaystyle M}$ ${\displaystyle (U_{i},\kappa _{i})_{i\in I}}$ ${\displaystyle \operatorname {supp} (\mu )}$ ${\displaystyle (\phi _{i})_{i\in I}}$

${\displaystyle \int _{M}\mu :=\sum _{i}\int _{\kappa _{i}(U)}\kappa _{i}^{*}(\phi _{i}\mu )}$.

The right side is independent of the choice of card and the choice of decomposition of the one.

properties

• The integral is invariant with respect to diffeomorphisms . That is, for all smooth manifold and the same dimension and each diffeomorphism and each 1-density applies ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle \phi \colon M\to N}$ ${\displaystyle \mu \in \Gamma _{c}^{\infty }(|\Lambda ^{n}(N)|^{1})}$
  ${\displaystyle \int _{M}\phi ^{*}\mu =\int _{N}\mu .}$
• The integral is local, that is, for each subset , and each 1-density with applicable ${\displaystyle U\subset M}$ ${\displaystyle \mu \in \Gamma _{c}^{\infty }(|\Lambda ^{n}(M)|^{1})}$ ${\displaystyle \operatorname {supp} (\mu )\subset U}$
  ${\displaystyle \int _{M}\mu =\int _{U}\mu .}$
• The following applies to each: The right integral is a normal Lebesgue integral of a smooth function with a compact support. ${\displaystyle \rho \in C_{c}^{\infty }(\mathbb {R} ^{n})}$
  ${\displaystyle \int _{\mathbb {R} ^{n}}\rho |\mathrm {d} v|=\int _{\mathbb {R} ^{n}}\rho \mathrm {d} x.}$
  

L ¹ space

Be a measurable 1-density with a compact carrier. If the integral exists , a -cut is called its norm through ${\displaystyle \mu \in |\Lambda ^{n}(M)|^{1}}$ ${\displaystyle \textstyle \int _{M}|\mu |}$ ${\displaystyle \mu }$ ${\displaystyle L^{1}}$

${\displaystyle \|\mu \|_{L^{1}}:=\int _{M}|\mu |}$

given is. The completion of this set with respect to the given norm gives the space. If the manifold is compact, the completion does nothing. ${\displaystyle L^{1}(M,|\Lambda ^{n}(M)|^{1}).}$

L ^p spaces

Now be and and one of the two densities have compact support. Then due to the property two is from the dual space section and has compact support. Thus it can be integrated. ${\displaystyle \mu \in |\Lambda ^{n}(M)|^{r}}$ ${\displaystyle \nu \in |\Lambda ^{n}(M)|^{1-r}}$ ${\displaystyle \mu \otimes \nu =\mu \cdot \nu \in |\Lambda ^{n}(M)|^{1}}$ ${\displaystyle \mu \cdot \nu }$

If it can be integrated, one speaks of a -cut through its norm ${\displaystyle \int _{M}|\mu |^{p}}$ ${\displaystyle L^{p}}$

${\displaystyle \|\mu \|_{L^{p}}:=\int _{M}|\mu |^{p}}$

given is. The completion provides the space Also again because of property two from the section dual space is the space with the dual space too ${\displaystyle L^{p}(M,|\Lambda ^{n}(M)|^{\frac {1}{p}}).}$ ${\displaystyle L^{q}(M,|\Lambda ^{n}(M)|^{\frac {1}{q}})}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle L^{p}(M,|\Lambda ^{n}(M)|^{\frac {1}{p}}).}$

Examples

Density bundle over real space

Let be the manifold to be considered. The tangential bundle is a trivial vector bundle , therefore global sections exist in and in the density bundle . Let be the canonical basis of , then is a basis of one-dimensional space . There is then a smooth nowhere cut that goes through ${\displaystyle M=\mathbb {R} ^{n}}$ ${\displaystyle T\mathbb {R} ^{n}}$ ${\displaystyle T\mathbb {R} ^{n}}$ ${\displaystyle \Lambda ^{n}(\mathbb {R} ^{n})}$ ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle e_{1}\wedge \cdots \wedge e_{n}}$ ${\displaystyle \Lambda ^{n}(\mathbb {R} ^{n})}$ ${\displaystyle \textstyle |\mathrm {d} \nu _{n}|\in |\Lambda ^{n}(\mathbb {R} ^{n})|^{1}}$

${\displaystyle |\mathrm {d} \nu _{n}|(e_{1}\wedge \cdots \wedge e_{n})=1}$

is defined. For every smooth map there is a smooth 1 density. The object can be understood as the Lebesgue measure . ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \mu =f|\mathrm {d} \nu _{n}|}$ ${\displaystyle |\mathrm {d} \nu _{n}|}$

Let be a smooth diffeomorphism then ${\displaystyle \phi =(\phi _{1},\ldots ,\phi _{n})\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$

${\displaystyle \phi ^{*}(|\mathrm {d} \nu _{n}|)=\left|\det \left({\frac {\partial \phi _{i}}{\partial x^{j}}}\right)\right|\cdot |\mathrm {d} \nu _{n}|.}$

The Jacobi matrix denotes by . This relationship can also be found in the coordinate transformation of integrals. Compare also transformation theorem . ${\displaystyle \textstyle \left({\frac {\partial \phi _{i}}{\partial x^{j}}}\right)}$ ${\displaystyle \phi }$

Riemann density

Let be an n-dimensional Riemannian manifold , then an orthonormal frame exists for the tangential bundle with respect to the Riemannian metric. The clearly determined global cut with ${\displaystyle (M,g)}$ ${\displaystyle e_{1},\ldots e_{n}}$ ${\displaystyle |\mathrm {d} x|\in \Gamma (M,|\Lambda ^{n}(M)|)}$

${\displaystyle |\mathrm {d} x|(e_{1}\wedge \cdots \wedge e_{n})=1}$

is called Riemann density. This cut always exists without further requirements.

Tensor density

→ Main article : Tensor density

In the definition of, replace the tangential bundle with the tensor bundle. Then the density bundle induced by it is called the -tensor density bundle . In this case the elements are called tensor fields . ${\displaystyle |\Lambda ^{n}(M)|^{r}:=|\Lambda ^{n}(TM)|^{r}}$ ${\displaystyle TM}$ ${\displaystyle T_{s}^{r}(TM).}$ ${\displaystyle |\Lambda ^{n}(T_{s}^{r}(TM))|^{r}}$ ${\displaystyle r}$ ${\displaystyle r=0}$

Distributions

Since one can integrate 1-densities over subsets of a manifold, as described above in the article, this now allows distributions to be defined on manifolds. Be the space of smooth cuts with a compact carrier . So you can get a distribution induced by ${\displaystyle \Gamma _{c}^{\infty }(|\Lambda ^{n}(M)|^{1})}$ ${\displaystyle M\to |\Lambda ^{n}(M)|^{1}}$ ${\displaystyle f}$

${\displaystyle T_{f}\colon \Gamma _{c}^{\infty }(|\Lambda ^{n}(M)|^{1})\to C^{\infty }(\mathbb {R} )}$

define by

${\displaystyle \omega \mapsto \int f\omega .}$

For this reason you bet

${\displaystyle {\mathcal {D}}(M,|\Lambda ^{n}(M)|^{1}):=\Gamma _{c}^{\infty }(|\Lambda ^{n}(M)|^{1}).}$

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

${\displaystyle {\mathcal {D}}'(M):=({\mathcal {D}}(M,|\Lambda ^{n}(M)|^{1}))'.}$

literature

• Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore et al. 2007, ISBN 978-981-270-853-3 .
• SR Simanca: Pseudo-differential operators (= Pitman Research Notes in Mathematics Series 236). Longman Scientific & Technical et al., Harlow et al. 1990, ISBN 0-582-06693-X .

Individual evidence

1. ^ ^{A b} Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore et al. 2007, ISBN 978-981-270-853-3 , p. 108.
2. ^ Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , p. 33.