1 Theory

1.0 Introduction

The package DifferentialForms (in file dform.spad) builds on the domain DeRhamComplex. In the following section we give a brief overview of the functions that are going to be implemented. The focus is on precise definitions of the notions, since those may be varying in the literature. In section (2) we will describe the exported functions and how they work, in section (3) some short implementation notes will be given and finally the last section is devoted to some examples.

1.1 Definitions

Let \(\mathcal{M}\) be a n-dimensional manifold (sufficiently smooth and orientable). To each point \(P \in \mathcal{M}\) there is a neighborhood which can be diffeomorphically mapped to some region in \(\mathbb{R}^n\), with coordinates

\[x_1 (P'), \ldots, x_n (P')\]

for all \(P' \in \mathcal{U} (P) \subset \mathcal{M}\). The tangent space \(T_{P'} (\mathcal{M})\) at the point \(P'\) is a vector space, that is spanned by the basis

\[e_1 (P'), \ldots, e_n (P')\]

which also is often denoted by

\[\partial_1, \ldots, \partial_n = \frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}.\]

A tangent vector \(v\) has the form

\[v = \sum_{j = 1}^n v^j e_j .\]

The cotangent space \(T_{P'}^{} (\mathcal{M})^{\star}\) is the vector space of linear functionals

\[\alpha : T_{P'} (\mathcal{M}) \rightarrow \mathbb{R},\]

spanned by the basis \(e^1 (P'), \ldots, e^n (P')\) which (corresponding to the basis \(\partial_j\)) is also denoted by

\[d x^1,\ldots, d x^n.\]

The latter notation indicates the dependency on the moving point \(P'\). The dual basis is by definition comprised of those linear functionals such that

\[e^j (e_k) = \delta^j_k .\]

Therefore we have

\[\alpha (v) = \alpha \left( \sum_{j = 1}^n v^j e_j \right) = \sum_{j = 1}^n v^j \alpha (e_j) = \sum_{j = 1}^n v^j \alpha_j,\]

where \(\alpha = \sum_{j = 1}^n \alpha_j e^j\).

1.1.1 Inner product of differential forms (dot)

Let \(g_x\) be a symmetric \(n \times n\) matrix which is nondegenerate (i.e. \(\det (g_x) \neq 0\)). The index x indicates that this matrix depends on the coordinates \(x_1 (P), \ldots, x_n (P)\) and may be varying from point to point. If this dependency is smooth (enough) we speak of a (pseudo-) Riemannian metric (locally). This way we get an isomorphism between tangent vectors and 1-forms (= covectors):

\[\alpha_j = g_{j k} v^k, \hspace{1.2em} v^j = g^{j k} \alpha_j .\]

Clearly, \(\sum_k g^{j k} g_{k l} = \delta^j_l\), in other words \((g^{j k})\) is the inverse of g. The metric g defines an inner product of vectors

\[g (v, w) = \langle v, w \rangle : = g_{i j} v^i w^j\]

and by duality also on 1-forms:

\[g^{- 1} (\alpha, \beta) = \langle \alpha, \beta \rangle : = g^{i j} \alpha_i \beta_j .\]

Now, this inner product is extended to arbitrary p-forms by

\[\langle \alpha_1 \wedge \ldots \wedge \alpha_p , \beta_1 \wedge \ldots \wedge \beta_p \rangle : = \det (\langle \alpha_i, \beta_j \rangle) , \hspace{1.8em} (1 \leqslant i, j \leqslant p), \label{dot}\]

and linearity.

1.1.2. The volume form \(\eta\) (volumeForm)

The Riemannian volume form \(\eta\) is (by definition) given by the n-form

\[\eta = \sqrt{| \det g |} e^1 \wedge \ldots \wedge e^n = \sqrt{| \det\,g |} d x^1 \wedge \ldots \wedge d x^n . \label{vol}\]

This definition makes sense because by a (orientation preserving) change of coordinates \(\sqrt{\mathrm{det} g}\) transforms like the component of a n-form.

1.1.3. Hodge dual (hodgeStar)

The Hodge dual of a differential p-form \(\beta\) is the (n - p)-form \(\star \beta\) such that

\[\alpha \wedge \star \beta = \langle \alpha, \beta \rangle \eta \label{hodge}\]

holds, for all p-forms \(\alpha\). The linear operator \((\star)\) is called the Hodge star operator. By the Riesz representation theorem the Hodge dual is uniquely defined by the expression above.

Warning

Flanders [4] defines the Hodge dual by the equality

\[\lambda \wedge \mu = \langle \star \lambda, \mu \rangle \eta\]

where \(\lambda\) is a p-form and \(\mu\) a (n - p)-form. This can result in different signs (actually \(\star_F = s(g)\star\), where \(s(g)\) is the sign of the determinant of \(g\)).

The generally adopted definition (2016) is the one given at the beginning of this subsection.

The components of \(\star \beta\) are

\[(\star \beta)_{j_1, \ldots, j_{n - p}} = \frac{1}{p!} \varepsilon_{i_1, \ldots, i_p, j_1, \ldots, j_{n - p}} \sqrt{| \det g |} g^{i_1 k_1} \ldots g^{i_p k_p} \beta_{k_1, \ldots, k_p}\]

what is equal to

\[\frac{1}{p! \sqrt{| \det g |}} \varepsilon_{}^{k_1, \ldots, k_p, l_1, \ldots, l_{n - p}} g_{j_1 l_1} \ldots g_{j_{n - p}, l_{n - p}} \beta_{k_1, \ldots, k_p} .\]

1.1.4 Interior product (interiorProduct)

The interior product of a vectorfield \(v\) and a p-form \(\alpha\) is a (p -1)-form \(i_v (\alpha)\) such that

\[i_v (\alpha) (v_1, \ldots, v_{p - 1}) = \alpha (v, v_1, \ldots, v_{p - 1})\]

holds, for all vectorfields \(v_1, \ldots, v_{p - 1}\). Therefore, the components of \(i_v (\alpha)\) are

\[i_v (\alpha)_{j_1, \ldots, j_{p - 1}} = v^j \alpha_{j, j_1, \ldots, j_{p - 1} .}\]

One can express the interior product by using the \(\star\)-operator. Let \(\alpha\) be the 1-form defined by the equation

\[\alpha (w) = g (v, w), \forall w.\]

That means in components: \(\alpha_j = g_{j k} v^k\), thus we have

\[i_v (\beta) = (-)^{p - 1} \star^{- 1} (\alpha \wedge \star \beta) .\]

Clearly, the interior product is independent of any metric, whereas the Hodge operator is not! So, usually one should not use the Hodge operator to compute the interior product.

We will use the fact that the interior product is an antiderivation, which allows a recursive implementation.

1.1.5 The Lie derivative (lieDerivative)

The Lie derivative with respect to a vector field \(v\) can be calculated (and defined) using Cartan’s formula:

\[\mathcal{L}_v \alpha = d i_v (\alpha) + i_v (d \alpha).\]

There are other ways to define \(\mathcal{L}_v \alpha\), however, it is convenient to compute it this way when \(d\) and \(i_v\) are already at hand.

1.1.6 The CoDifferential \(\delta\) (codifferential)

The codifferential \(\delta\) is defined on a p-form as follows:

\[\delta = (-1)^{n(p-1)+1}\,s(g) \star\,d\,\star\]

where \(g\) is the metric and \(s(g)\) is related to the signature of \(s(g)\) as described next.

1.1.7 The sign of a metric \(s(g)\) (s)

The signature of a metric \(g\) is defined as the difference of the number of positive (p) and negative (q) eigenvalues, i.e:

\[\mathrm{signature(g)} = p - q\]

and the sign functions s is defined as:

\[s(g) = (-1)^{\frac{n -\mathrm{signature(g)}}{2}}\]

Since, as we always assume, \(g\) is non-degenerate, we have \(p+q=n\), and consequently:

\[s(g) = (-1)^{q} = \mathrm{sign}\, \mathrm{det}(g)\]

1.1.8 The inverse Hodge star \(\star^{-1}\) (invHodgeStar)

Applying the Hodge star operator twice on a p-form twice we get the identity up to sign:

\[\star\circ\star\, \omega_p = (-1)^{p(n-p)}\,s(g)\,\omega_p.\]

Therefore

\[\star^{-1}\,\omega_p = (-1)^{p(n-p)}\,s(g)\,\star\omega_p.\]

1.1.9 The Hodge-Laplacian \(\Delta_g\) (hodgeLaplacian)

The Hodge-Laplacian also known as Laplace-de Rham operator is defined on any manifold equipped with a (pseudo-) Riemannian metric \(g\) and is given by

\[\Delta_g = d\circ\delta + \delta\circ d\]

Note that in the Euclidean case \(\Delta_g = - \Delta\), where latter is the ordinary Laplacian.

Bibliography

References

[1]	Ralph Abraham, Jerrold E.Marsden and Tudor Ratiu.Manifolds, Tensor Analysis, and Applications. Springer, Auflage: 2nd Corrected ed. 1988. Corr. 2nd printing 1993 edition.

[2]	Henri Cartan. Di erential Forms. Dover Pubn Inc, Au age: Tra edition.

[3]	Herbert Federer. Geometric Measure Theory. Springer, Au age: Reprint of the 1st ed. Berlin, Heidelberg, New York 1969 edition.

[4]	Harley Flanders and Mathematics. Differential Forms with Applications to the Physical Sciences. Dover Pubn Inc, Auflage: Revised. edition.

[5]	L. A. Lambe and D. E. Radford. Introduction to the Quantum Yang-Baxter Equation and Quantum Groups:An Algebraic Approach. Springer, Auflage: 1997 edition.

[6]	Walter Rudin and RudinWalter. Principles ofMathematicalAnalysis.Mcgraw Hill Book Co, Au age: Revised. edition.

[7]	Hassler Whitney. Geometric Integration Theory: Princeton Mathematical Series, No. 21. Literary Licensing, LLC.