Hodge star operator

The Hodge star operator or Hodge operator for short is an object from differential geometry . It was introduced by the British mathematician William Vallance Douglas Hodge . The operator is an isomorphism which operates on the outer algebra of a finite-dimensional Prehilbert space or, more generally, on the space of differential forms .

motivation

Let be an n-dimensional, smooth manifold and let be the -th external power of the cotangent space . For all with the vector spaces and have the same dimension and are therefore isomorphic. If , in addition, the structure of an oriented , semiriemannian manifold has been added , one can prove that this isomorphism can be constructed naturally. That is, there is an isomorphism between the spaces that is invariant under the Semiriemannian metric and the orientation-preserving diffeomorphisms . The generalization of this isomorphism to the tangential bundle is called the Hodge-Star operator. ${\displaystyle M}$ ${\displaystyle \Lambda ^{k}T_{p}^{*}M}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle 0\leq k\leq n}$ ${\displaystyle \Lambda ^{k}T_{p}^{*}M}$ ${\displaystyle \Lambda ^{n-k}T_{p}^{*}M}$ ${\displaystyle M}$

definition

Since the space is a finite-dimensional vector space from the above motivation, we begin here with the definition of the Hodge-Star operator on vector spaces. ${\displaystyle T_{p}^{*}M}$

Hodge star operator on vector spaces

Let be a -dimensional oriented vector space with scalar product and its dual space . For , the -th external power of denotes the vector space of the alternating multilinear forms of the level above . ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle V^{*}}$ ${\displaystyle 0\leq k\leq n}$ ${\displaystyle \Lambda ^{k}(V^{*})}$ ${\displaystyle k}$ ${\displaystyle V^{*}}$ ${\displaystyle k}$ ${\displaystyle V}$

The Hodge star operator

${\displaystyle \ast \colon \Lambda ^{k}(V^{*})\rightarrow \Lambda ^{n-k}(V^{*})}$

is clearly defined by the following condition: If there is a positively oriented orthonormal basis of and the corresponding dual basis of , then is ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ ${\displaystyle V}$ ${\displaystyle \{e^{1},\ldots ,e^{n}\}}$ ${\displaystyle V^{*}}$

${\displaystyle *(e^{1}\wedge e^{2}\wedge \dots \wedge e^{k})=e^{k+1}\wedge e^{k+2}\wedge \dots \wedge e^{n}}$

It is not enough to require this condition for a single orthonormal basis. But one does not need to demand it for every positively oriented orthonormal basis. It suffices to consider all even permutations of a single basis: If there is a positively oriented orthonormal basis of and the dual basis of it is , then the Hodge-Star operator is uniquely determined by the condition ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ ${\displaystyle V}$ ${\displaystyle \{e^{1},\ldots ,e^{n}\}}$ ${\displaystyle V^{*}}$

${\displaystyle \ast (e^{\sigma (1)}\wedge e^{\sigma (2)}\wedge \cdots \wedge e^{\sigma (k)})=e^{\sigma (k+1)}\wedge e^{\sigma (k+2)}\wedge \cdots \wedge e^{\sigma (n)}}$

for every even permutation of . ${\displaystyle \sigma }$ ${\displaystyle \{1,\dots ,n\}}$

For an orthogonal basis, which need not be an orthonormal basis, more general applies

${\displaystyle \ast (e_{\sigma (1)}\wedge \ldots \wedge e_{\sigma (k)})=g_{\sigma (1)\sigma (1)}\ldots g_{\sigma (k)\sigma (k)}{\frac {s(B)}{\sqrt {|\det g|}}}\,\mathrm {sgn} (\sigma )\,e_{\sigma (k+1)}\wedge \ldots \wedge e_{\sigma (n)}}$

and

${\displaystyle \ast (e^{\sigma (1)}\wedge \ldots \wedge e^{\sigma (k)})=g^{\sigma (1)\sigma (1)}\ldots g^{\sigma (k)\sigma (k)}s(B){\sqrt {|\det g|}}\,\mathrm {sgn} (\sigma )\,e^{\sigma (k+1)}\wedge \ldots \wedge e^{\sigma (n)}}$.

Here is when is positively oriented and when is negatively oriented. The formula is especially valid for empty products, so for an orthonormal basis is ${\displaystyle s(B)=1}$ ${\displaystyle B}$ ${\displaystyle s(B)=-1}$ ${\displaystyle B}$

${\displaystyle \ast 1=s(B)\,e_{1}\wedge \ldots \wedge e_{n}}$,

${\displaystyle \ast (e_{1}\wedge \ldots \wedge e_{n})=s(B)}$.

Global Hodge star operator

After this preliminary work, the Hodge-Star operator can be transferred to the outer algebra of the cotangent bundle . As in motivation, let it be an orientable, smooth Riemannian manifold. Also define as the space of the cuts in the vector bundle . The space is therefore the space of the differential forms -th degree . Since there is a vector bundle and therefore there is a vector space in every point , the Hodge star operator is defined point by point. ${\displaystyle \textstyle T^{*}M=\bigsqcup _{p\in M}T_{p}^{*}M}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {A}}^{k}(M)}$ ${\displaystyle \Lambda ^{k}(T^{*}M)}$ ${\displaystyle {\mathcal {A}}^{k}(M)}$ ${\displaystyle k}$ ${\displaystyle M}$ ${\displaystyle T^{*}M}$ ${\displaystyle p\in M}$

The Hodge star operator is an isomorphism

${\displaystyle {\begin{aligned}\ast :{\mathcal {A}}^{k}(M)&\to {\mathcal {A}}^{n-k}(M)\\\omega &\mapsto \ast \omega ,\end{aligned}}}$

so that for each point ${\displaystyle p\in M}$

${\displaystyle \omega |_{p}\mapsto \ast \omega |_{p}}$

applies. The differential form , evaluated at this point , is again an element of a vector space, and thus the above definition for vector spaces applies. In this definition it was implied that the shape is again a smooth differential shape . However, this is not clear and needs to be proven. ${\displaystyle \omega }$ ${\displaystyle p}$ ${\displaystyle \ast \omega }$

Examples

If one considers the three-dimensional Euclidean space as a Riemannian manifold with the Euclidean metric and the usual orientation, then one can apply the Hodge star operator under these conditions. Let be the oriented standard basis of and the corresponding dual basis . The elements can then be understood as differential forms. The following then applies to the Hodge star operator ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \{e_{x},e_{y},e_{z}\}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \{\mathrm {d} x,\mathrm {d} y,\mathrm {d} z\}}$ ${\displaystyle \mathrm {d} x,\mathrm {d} y,\mathrm {d} z}$ ${\displaystyle \ast }$

${\displaystyle {\begin{aligned}\ast \mathrm {d} x&=\mathrm {d} y\wedge \mathrm {d} z\\\ast \mathrm {d} y&=\mathrm {d} z\wedge \mathrm {d} x\\\ast \mathrm {d} z&=\mathrm {d} x\wedge \mathrm {d} y\,.\end{aligned}}}$

Under these conditions, the Hodge star operator is used implicitly in vector analysis for the cross product and the rotation operator derived from it . This is explained in the article External Algebra .

Properties of the Hodge star operator

Be an oriented smooth Riemannian manifold are, , , and is a Riemannian metric . Then the Hodge star operator has the following properties: ${\displaystyle M}$ ${\displaystyle f_{1},f_{2}\in C^{\infty }(M)}$ ${\displaystyle \omega ,\nu \in {\mathcal {A}}^{k}(M)}$ ${\displaystyle g(\cdot ,\cdot )}$

1. ${\displaystyle \ast (f_{1}\omega +f_{2}\nu )=f_{1}\cdot \ast \omega +f_{2}\cdot \ast \nu }$ (Linearity),
2. ${\displaystyle \ast \ast \omega =(-1)^{k(n-k)}\cdot \omega }$ (Bijectivity),
3. ${\displaystyle \omega \wedge *\nu =\nu \wedge *\omega =g(\omega ,\nu )\cdot \ast 1_{M},}$
4. ${\displaystyle \ast (\omega \wedge \ast \nu )=\ast (\nu \wedge \ast \omega )=g(\omega ,\nu ),}$
5. ${\displaystyle g(\ast \omega ,\ast \nu )=g(\omega ,\nu )}$( Isometry ).

Riemann volume form

Be a smooth, oriented, Riemannian manifold. If one then understands as a constant one function, the Riemann volume shape is defined as . This volume shape is an important part of the integration with differential shapes. This should be illustrated with a simple example. For this, be a compact subset. For the volume of U applies . If one understands now as a manifold and as a compact subset contained therein, then the volume in this case is defined as ${\displaystyle M}$ ${\displaystyle 1\in C^{\infty }(M)={\mathcal {A}}^{0}(M)}$ ${\displaystyle *1}$ ${\displaystyle U\subset \mathbb {R} ^{3}}$ ${\displaystyle \textstyle \mathrm {Vol} (U)=\int _{U}1\ \mathrm {d} (x_{1},x_{2},x_{3})}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle U}$

${\displaystyle \operatorname {Vol} (U):=\int _{U}*1=\int _{U}\mathrm {d} x_{1}\wedge \mathrm {d} x_{2}\wedge \mathrm {d} x_{3}=\int _{U}1\mathrm {d} (x_{1},x_{2},x_{3}).}$

The integration theory on manifolds also includes integration on real subsets. According to this principle one can also integrate functions on manifolds by multiplying them with the volume form.

literature

• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications . Springer-Verlag, Berlin 2003, ISBN 3-540-96790-7 .
• S. Morita: Geometry of Differential Forms . American Mathematical Society, ISBN 0-821-81045-6 .