nLab differential form

Idea

A differential form is a geometrical object on a manifold that can be integrated. A differential form $\omega$ is a section of the exterior algebra $\Lambda^* T^* X$ of a cotangent bundle, which makes sense in many contexts (e.g. manifolds, algebraic varieties, analytic spaces, …).

What we're actually describing here are the exterior differential forms; for more general concepts, see absolute differential form and cogerm differential form.

Definitions

Standard definition

Given a differentiable manifold $X$, or even a generalized smooth space $X$ for which this definition makes sense, a differential form on $X$ is a section of the exterior algebra of the cotangent bundle over $X$; sometimes one refers to an exterior differential form to be more precise. One often requires differential forms to be smooth, or at least continuous, but we will state this explicitly when we want it. A differential $p$-form on $X$ is a section of the $p$th exterior power of the cotangent bundle; the natural number $p \geq 0$ is the rank of the form. A general differential form is a $p$-indexed sequence of differential $p$-forms of which all but finitely many are zero; on a finite-dimensional manifold, this latter condition is automatic.

The space $C^\infty\Omega^*(X)$ of smooth forms on $X$ may also be defined as the universal differential envelope of the space $C^\infty\Omega^0(X)$ of smooth functions on $X$ (which are the same as the smooth $0$-forms as defined above); more concretely:

It is generated by the smooth functions and three operations:

• an associative binary operation of addition generalising the usual addition of functions,
• an associative binary operation $\wedge$ (the exterior product or wedge product) generalising the usual multiplication of functions,
• a unary operation $\mathrm{d}$ (the exterior derivative, or differential);

subject to these identities:

• addition makes $C^\infty\Omega^*(X)$ into an abelian monoid,
• both $\mathrm{d}$ and $\wedge$ distribute over addition,
• $\mathrm{d}\mathrm{d}f = 0$,
• $\mathrm{d}f \wedge \mathrm{d}f = 0$,
• $\mathrm{d}(f \wedge \eta) = \mathrm{d}f \wedge \eta + f \wedge \mathrm{d}\eta$,
• $\mathrm{d}(\mathrm{d}f \wedge \eta) + \mathrm{d}f \wedge \mathrm{d}\eta = 0$,

in which $f$ is a smooth function and $\eta$ is an arbitrary smooth form. (Note that one often drops the ‘$\wedge$’ after a $0$-form; thus, $f \eta = f \wedge \eta$. There is hardly any ambiguity if one drops the ‘$\wedge$’ entirely, but it's traditional.)

Although not directly stated, it can be proved that addition makes $C^\infty\Omega^*(X)$ into an abelian group; in fact, it is a module of the commutative ring of smooth functions on $X$. This is further a graded module, graded by the natural numbers, with the elements of grade $p$ being the $p$-forms defined earlier; the space of these is $C^\infty\Omega^p(X)$. If $\omega$ is a $p$-form and $\eta$ is a $q$-form, we have:

• $\mathrm{d}\omega$ is a $(p+1)$-form,
• $\omega \wedge \eta$ is a $(p+q)$-form.

The law

$$\mathrm{d}\mathrm{d}\eta = 0$$

holds for any form $\eta$, but the other laws become more complicated; if $\omega$ is a $p$-form and $\eta$ is a $q$-form, then we get

• $\omega \wedge \omega = 0$ if $p$ is odd,
• $\omega \wedge \eta = (-1)^{pq}\, \eta \wedge \omega$,
• $\mathrm{d}(\omega \wedge \eta) = \mathrm{d}\omega \wedge \eta + (-1)^p\, \omega \wedge \mathrm{d}\eta$.

That is, $C^\infty\Omega^*(X)$ is a skew-commutative algebra over the ring of smooth functions, equipped with a derivation $\mathrm{d}$ of degree $1$. In fact, the description above in terms of generators and relations makes it the free skew-commutative algebra over that ring equipped with such a derivation. (Or if it doesn't, then it's because I left something out of that description.)

More general forms (in $\Omega^*(X)$) can be recovered as sums of terms, each of which is the wedge product of a function and a smooth form. (This can also be seen as a special case of a vector-valued form as below.) One can still define the exterior derivative of a $C^1$ (once continuously differentiable) form; in general, the differential of a $C^k$ form is a $C^{k-1}$ form. If $X$ is not a smooth manifold but only $C^k$ for some $1 \leq k \lt \infty$, then one has to take more care here, but the definition of the skew-commutative algebra of $C^k$ differential forms can still be made to work.

Given local coordinates $(x^1, \ldots, x^n)$ on a patch $U$ in an $n$-dimensional manifold $X$, any differential form $\eta$ on $U$ can be expressed uniquely as a sum of $2^n$ terms

$$\eta = \sum_I \eta_I \wedge \mathrm{d}x^I ,$$

where $I$ runs over increasing lists of indices from $(1,\ldots,n)$, each $\eta_I$ is a function on $U$ (continuous, smooth, etc according as $\eta$ is), and

$$\mathrm{d}x^I = \mathrm{d}x^{I_1} \wedge \cdots \wedge \mathrm{d}x^{I_p}$$

(for $p$ the length of the list $I$) is simply an abbreviation. For a $p$-form, there are $\left(n \atop p\right)$ terms that appear.

Twisted and vector-valued forms

Recall that a differential form on $X$ is a section of the exterior algebra of the cotangent bundle over $X$; call this bundle $\Lambda$. Then given any vector bundle $V$ over $X$, a $V$-valued form on $X$ is a section of the vector bundle $V \otimes \Lambda$. The wedge product of a $V$-valued form and a $V'$-valued form is a $(V \otimes V')$-valued form, but if there is a commonly used multiplication map $V \otimes V' \to W$, then we may think of their wedge product as a $W$-valued form.

Of particular importance are $L$-valued forms when $L$ is a line bundle; these are also called $L$-twisted forms. (Compare the notion of twisted form in a more general context.) In local coordinates, a twisted form looks just like an ordinary form, once you choose a nonzero vector in $L$ as a basis. Therefore, they can seem sneaky and confusing sometimes when you realise that they do not behave in the same way!

Let $\Psi$ be the pseudoscalar bundle, obtained as the bundle associated to the frame bundle by the character $\rho(A)=sgn(det(A))$. That is, a section of $\Psi$ (a pseudoscalar field) is given locally by a simple scalar field (a real-valued function) for each orientation of a local patch, with opposite orientations giving oppositely-signed scalars. A $\Psi$-twisted form is called pseudoform.

On an $n$-dimensional manifold $X$, the bundle $\Lambda^n(X)=\Lambda^n(T^*M)$ is itself a line bundle, which is associated to the frame bundle by the character $A\mapsto det(A)^{-1}$; a $p$-form twisted by this line bundle is called a densitised form. Sometimes an $n$-form is itself called a density. As we will see under integration below, it is really an $n$-pseudoform (a section of the bundle $\Lambda^n\otimes\Psi$, which comes from the character $A\mapsto |\det(A)|^{-1}$) that should be called a density, but that is not the traditional terminology.

Given any real number $w$, the bundle $|\Lambda|^w$ associated to the character $A\mapsto |det A|^{-w}$ is called the line bundle of $w$-weighted scalars (also called $w$-densities); a form twisted by this line bundle is a $w$-weighted form. Note that $|\Lambda|^0=1$ is the trivial bundle, so a $0$-weighted form is just an ordinary form. Also, an $n$-pseudoform turns out to be equivalent to a $1$-weighted scalar:

$$\Lambda^n \otimes \Psi \simeq |\Lambda|^1.$$

Since $\Psi\otimes\Psi=1$ is the trivial bundle, we have $\Lambda^n \simeq |\Lambda|^1 \otimes\Psi$, and thus a densitised form is equivalent to a $1$-weighted pseudoform:

$$\Lambda^k \otimes \Lambda^n \simeq \Lambda^k \otimes \Psi \otimes |\Lambda|^1$$

The bundle $\Lambda^n\otimes\Psi\simeq|\Lambda|^1$ of $n$-pseudoforms is the absolute value of the bundle of $n$-forms (that is of densitised $0$-forms), so we may take the absolute value of one of either and get one of the latter. Similarly, the bundle of $0$-forms is the absolute value of the bundle $\Psi$ of pseudo-$0$-forms; that is, the trivial bundle is the absolute value of the pseudoscalar bundle.

As smooth functors on $n$-paths

One way to exhibit this statement nicely is:

A differential $n$-form on $X$ is a smooth $n$-functor $P_n(X) \to \mathbf{B}^n \mathbb{R}$ from the path n-groupoid of $X$ to the $n$-fold delooping of the additive Lie group of real numbers.

Urs, do you know where the need for orientation comes in here? I don't follow it in enough detail to see, although I intend to read Moerdijk–Reyes. —Toby

Eric: I’m probably confused, but if $\sigma_n$ is a morphism in $P_n(X)$, then (unless $X$ is a directed space), the opposite $\sigma_n^{-1}$ is also in $P_n(X)$ and I think $\omega(\sigma_n) = -\omega(\sigma_n^{-1})$.

Toby: Between Eric's comment here and Urs's at latest changes?, I'm happy to remove this query box.

Eric: I think it is a good question. Maybe we should keep the query box here until the answer is incorporated in the page.

Urs Schreiber: here is my reply, that I originally posted at latest changes?. I’ll try to eventually work this into the main text of the entry

The orientation of the diffential form corresponds to the inherent orientation of k-morphisms: as we identify the differential form with a smooth functor on the path n-groupoid, that path n-groupoid necessarily has oriented $k$-volumes as its k-morphisms, simply because these $k$-morphisms need to come with information about their (higher categorical) source and target.

To get pseudo-differential forms that may be integrated also over non-oriented and possibly non-orientable manifolds one needs to consider parallel transport functors not with coefficients in just $\mathbf{B}^n \mathbb{R}$ coming from the crossed complex

$$(\mathbb{R} \to {*} \to \cdots \to {*})$$

but the more refined crossed complex

$$(\mathbb{R} \to {*} \to \cdots \to \mathbb{Z}_2)$$

where the $\mathbb{Z}_2$-factor acts by sign reversal on $\mathbb{R}$ (one can also use $U(1)$ instead of $\mathbb{R}$, this way $[P_n(-), \mathbf{B}^n U(1)]$ becomes the Deligne complex and knows not just about differential forms but about $U(1)$ $n-1$-gerbes with connection even).

A little bit of discussion of this unoriented case is currently at orientifold. There for the case $n=2$.

Note that an $n$-morphism in $P_n(X)$ is an oriented $n$-dimensional submanifold of $X$.

Such a functor (as described in more detail at connection on a bundle) assigns a real number to each parametrised $n$-dimensional cube of $X$, that is a subspace by a smooth map $\Sigma\colon [0,1]^n \to X$. If the differential form that this $n$-functor defines is denoted $\omega \in \Omega^n(X)$, then this real number is denoted by the integral

$$\int_{[0,1]^n} \Sigma^* \omega \,.$$

This integral in turn encodes the $n$-functoriality of the $n$-functor: it effectively says that

• if we decompose the standard $n$-cube $[0,1]^n$ into $N^n$ little subcubes $(C_k)_{k\in \mathbb{N}^n}$ for $N \in \mathbb{N}$
• and apply the $n$-functor to each of these to obtain a result (a real number) to be denoted $\omega(C_k)$;
• then by $n$-functoriality the result of the application of the functor to the full $\Sigma$ is the composition of all the $\omega(C_k)$ in $\mathbb{R}$. i.e. their sum
  $$\int_{[0,1]^n} \Sigma^* \omega = \sum_k \omega(C_k) \,.$$

Since one can let $N$ increase arbitrarily in this prescription – $N \to \infty$ – it follows that the value of the functor on $\Sigma$ is already determined by all its values on all “infinitesimal $n$-cubes” in some sense.

The notion of differential form is the one that makes this precise: a differential form is a rule for assigning to each “infinitesimal $n$-cube” a number.

There are in turn different ways to make that last statement precise:

• In differential geometry an “infinitesimal $n$-cube” is modeled by an $n$-tuple of tangent vectors and a differential form is a fiberwise linear map from the $n$-fold exterior power of the tangent bundle to the real numbers, as given below.

• In synthetic differential geometry the statement is in essence the same one, but the difference is that there the notion of “infinitesimal $n$-cube” has a concrete meaning on the same footing of other $n$-cubes. If $D^n$ denotes the abstract infinitesimal $n$-cube in this context, then the mapping space $X^{D^n}$ of morphisms from $D^n\to X$ is the $n$-fold tangent bundle of $X$ and a differential form is precisely nothing but a morphism

  $$\omega\colon X^{D^n} \times D^n \to \mathbb{R}$$

  (where $\mathbb{R}$ is now the synthetic differential version of the real numbers) subject to three constraints. (These constraints can be seen as the infinitesimal analog of the $n$-functoriality discussed above).

  This is described in detail in section 4 of

  • Moerdijk-Reyes, Models for Smooth Infinitesimal Analysis

  For more on this see differential forms in synthetic differential geometry.

More general kinds of forms

While differential forms are usually restricted to the linear case, there are also more general kinds of “differential forms” which can be integrated. See for instance absolute differential form and cogerm differential form.

Operations on differential forms

Pulling back forms

Given manifolds $X$ and $Y$ and a continuously differentiable map $f\colon X \to Y$, any differential form $\eta$ on $Y$ defines a pullback form $f^*(\eta)$ on $X$. See at pullback of differential forms.

Thus, the operation that maps $X$ to $\Omega^*(X)$ extends to a contravariant functor $\Omega^*$. Perhaps confusingly, forms are traditionally known in physics as ‘covariant’ concepts, because of how the components transform under a change of coordinates. (Ultimately, this confusion goes back to that between active and passive coordinate transformations.)

Note that twisted and (more general) vector-valued forms cannot be pulled back so easily. One needs some extra structure on $f$ to do so; see the discussion of integration of $p$-pseudoforms at integration of differential forms for an example.

Integration of forms

Let $X$ be an $n$-dimensional manifold, and let $\omega$ be an $n$-pseudoform on $X$. At least when $X$ is paracompact and Hausdorff, we may turn $\omega$ into a measure on $X$ and thereby find its integral. Conversely, any absolutely continuous Radon measure on $X$ arises in this way from a unique $n$-pseudoform $\omega$.

If we wish to integrate untwisted (or differently twisted or vector-valued) forms and/or forms of smaller rank, then we may do so on submanifolds of $X$ equipped with some appropriate structure. In particular, if $X$ itself is equipped with an orientation, then $n$-pseudoforms on $X$ are the same as (untwisted) $n$-forms, and so we can integrate those on $X$. See integration of differential forms for the general case.

Zoran Škoda: Should maybe this entry have a discussion on heuristics behind the usual trick in supersymmetry which asserts that the inner hom for supermanifolds, gives the statement that the algebra of smooth differential forms on $M$ is the space of functions on the odd tangent bundle $\Pi T M$? I am not the most competent to do this succinctly enough…

Toby: Possibly that should go at differential forms on supermanifolds?

Zoran Škoda: By no means. Ordinary differential forms on ORDINARY manifolds are the same as functions on odd tangent bundle. I did not want to say anything about the generalization of differential forms on supermanifolds. So it is NOT a different notion, but a different way to define it. If going to toposes hence synthetic framework is not separated why would be separated the equality which involves a parity trick…

Toby: Ah, I see; your $M$ above need not be super, and it still works. Then yes, that should be mentioned here too.

Cohomology of forms

There is a cohomology theory of smooth differential forms; we have a chain complex

$$\cdots \stackrel{\mathrm{d}}\to C^\infty\Omega^2(X) \stackrel{\mathrm{d}}\to C^\infty\Omega^1(X) \stackrel{\mathrm{d}}\to C^\infty\Omega^0(X) \stackrel{\mathrm{d}}\to 0 ;$$

the chain cohomology of this complex is the de Rham cohomology of $X$.

As smooth differential forms are the cochains in de Rham cohomolgy, the theory of integration of forms allows us to interpret relatively compact oriented submanifolds as chains on $X$, giving us a homology theory. Combining these, we have Stokes's theorem

$$\int_{\partial{R}} \omega = \int_R \mathrm{d}\omega ,$$

where $\partial{R}$, which may be interpreted as the boundary of $R$, is also called the codifferential as it is dual to $\mathrm{d}$.

Related concepts

• closed differential form, exact differential form

• wedge product

• decomposable differential form

• integration of differential forms

• horizontal differential form, vertical differential form

• harmonic differential form

• differential form on a supermanifold

• Kähler differential form

• de Rham complex, de Rham cohomology

• differential cohomology

• absolute differential form, cogerm differential form

• holomorphic differential form

• differential form with logarithmic singularities

References

General

The concept of differential forms (and their exterior algebra), at least on affine spaces/Euclidean spaces originates in

• Hermann Grassmann, Ausdehnungslehre, 1844

Textbook accounts:

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer (1982) [doi:10.1007/978-1-4757-3951-0]

• Georges de Rham, Chapter II of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]

• Manfredo P. Carmo, Differential Forms and Applications, Springer (1994) [doi:10.1007/978-3-642-57951-6]

• Victor Guillemin, Peter Haine, Differential Forms, World Scientific (2019) [doi:10.1142/11058, draft:pdf]

Exposition with eye towards applications in mathematical physics:

• Tohru Eguchi, Peter Gilkey, Andrew Hanson, Section 2 of: Gravitation, gauge theories and differential geometry, Physics Reports 66 6 (1980) 213-393 [doi:10.1016/0370-1573(80)90130-1]

• Theodore Frankel, section 2.1 of: The Geometry of Physics - An Introduction (1997)

An introductory wiki-format textbook is

• Tevian Dray, The Geometry of Differential Forms, web

published as half of

• Tevian Dray, Differential Forms and the Geometry of General Relativity, A K Peters/CRC Press, 2014, web.

The equivalence between differential forms and smooth functors on the path groupoid in low degree is discussed in

• Schreiber & Waldorf, Smooth Functors vs. Differential Forms (arXiv)

Much fun discussion between Eric Forgy, Toby Bartels, and John Baez, about whether integration of forms or pseudoforms is most fundamental (and about whether twisted forms in general are useful and interesting geometric objects or the bastard spawn of hell) may be found in this giant Usenet thread. More specifically:

• John's alter ego the Wizard explains why $n$-pseudoforms are the most natural things to integrate on an $n$-dimensional manifold;
• applications of pseudoforms to classical contexts where absolute values appear;
• the general notion of form corresponding to an arbitrary representation of the general linear group (towards the end of the post);
• absolute values of forms and how to integrate them (near the end of the post, a precursor to absolute differential forms).

Maxwell’s equations via differential forms

On the expression of classical electromagnetism, and especially of Maxwell's equations, in terms of differential forms, the de Rham differential and Hodge star operators:

• Élie Cartan, §80 in: Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite), Annales scientifiques de l’É.N.S. 3e série, tome 41 (1924) 1-25 [numdam:ASENS_1924_3_41__1_]

  (already in pregeometric form)

• Charles Misner, Kip Thorne, John Wheeler, §3.4 and §4.3 in: Gravitation, W. H. Freeman, San Francisco (1973) [ISBN:9780716703440]

• Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly 81 4 (1974) [doi:10.1080/00029890.1974.11993557, jstor:2318995, doi:10.2307/2318995]

• Walter Thirring, vol 2 §1.3 in: A Course in Mathematical Physics – 1 Classical Dynamical Systems and 2 Classical Field Theory, Springer (1978, 1992) [doi:10.1007/978-1-4684-0517-0]

• Theodore Frankel, §§9-10 in: Gravitational Curvature, Freeman, San Francisco (1979) [ark:13960/t58d7nn19]

• Dominic G. B. Edelen, §9.2 in: Applied exterior calculus, Wiley (1985) [GoogleBooks]

• Theodore Frankel, §3.5 & §7.2b in: The Geometry of Physics - An Introduction, Cambridge University Press (1997, 2004, 2012) [doi:10.1017/CBO9781139061377]

• Gregory L. Naber, §2.2 in: Topology, Geometry and Gauge fields – Interactions, Applied Mathematical Sciences 141 (2011) [doi:10.1007/978-1-4419-7895-0]

• Masao Kitano, Reformulation of Electromagnetism with Differential Forms, Chapter 2 in: Trends in Electromagnetism – From fundamentals to applications, InTech (2012) 21-44 [ISBN:978-953-51-0267-0, pdf]

• Sébastien Fumeron, Bertrand Berche, Fernando Moraes, Improving student understanding of electrodynamics: the case for differential forms, American Journal of Physics 88 (2020) 1083 [doi:10.1119/10.0001754, arXiv:2009.10356]