nLab differential form

Differential forms


Differential geometry

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Differential forms


A differential form is a geometrical object on a manifold that can be integrated. A differential form ω\omega is a section of the exterior algebra Λ *T *X\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.


Standard definition

Given a differentiable manifold XX, or even a generalized smooth space XX for which this definition makes sense, a differential form on XX is a section of the exterior algebra of the cotangent bundle over XX; 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 pp-form on XX is a section of the ppth exterior power of the cotangent bundle; the natural number p0p \geq 0 is the rank of the form. A general differential form is a pp-indexed sequence of differential pp-forms of which all but finitely many are zero; on a finite-dimensional manifold, this latter condition is automatic.

The space C Ω *(X)C^\infty\Omega^*(X) of smooth forms on XX may also be defined as the universal differential envelope of the space C Ω 0(X)C^\infty\Omega^0(X) of smooth functions on XX (which are the same as the smooth 00-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 d\mathrm{d} (the exterior derivative, or differential);

subject to these identities:

  • addition makes C Ω *(X)C^\infty\Omega^*(X) into an abelian monoid,
  • both d\mathrm{d} and \wedge distribute over addition,
  • ddf=0\mathrm{d}\mathrm{d}f = 0,
  • dfdf=0\mathrm{d}f \wedge \mathrm{d}f = 0,
  • d(fη)=dfη+fdη\mathrm{d}(f \wedge \eta) = \mathrm{d}f \wedge \eta + f \wedge \mathrm{d}\eta,
  • d(dfη)+dfdη=0\mathrm{d}(\mathrm{d}f \wedge \eta) + \mathrm{d}f \wedge \mathrm{d}\eta = 0,

in which ff is a smooth function and η\eta is an arbitrary smooth form. (Note that one often drops the ‘\wedge’ after a 00-form; thus, fη=fηf \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 Ω *(X)C^\infty\Omega^*(X) into an abelian group; in fact, it is a module of the commutative ring of smooth functions on XX. This is further a graded module, graded by the natural numbers, with the elements of grade pp being the pp-forms defined earlier; the space of these is C Ω p(X)C^\infty\Omega^p(X). If ω\omega is a pp-form and η\eta is a qq-form, we have:

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

The law

ddη=0\mathrm{d}\mathrm{d}\eta = 0

holds for any form η\eta, but the other laws become more complicated; if ω\omega is a pp-form and η\eta is a qq-form, then we get

  • ωω=0\omega \wedge \omega = 0 if pp is odd,
  • ωη=(1) pqηω\omega \wedge \eta = (-1)^{pq}\, \eta \wedge \omega,
  • d(ωη)=dωη+(1) pωdη\mathrm{d}(\omega \wedge \eta) = \mathrm{d}\omega \wedge \eta + (-1)^p\, \omega \wedge \mathrm{d}\eta.

That is, C Ω *(X)C^\infty\Omega^*(X) is a skew-commutative algebra over the ring of smooth functions, equipped with a derivation d\mathrm{d} of degree 11. 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 Ω *(X)\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 1C^1 (once continuously differentiable) form; in general, the differential of a C kC^k form is a C k1C^{k-1} form. If XX is not a smooth manifold but only C kC^k for some 1k<1 \leq k \lt \infty, then one has to take more care here, but the definition of the skew-commutative algebra of C kC^k differential forms can still be made to work.

Given local coordinates (x 1,,x n)(x^1, \ldots, x^n) on a patch UU in an nn-dimensional manifold XX, any differential form η\eta on UU can be expressed uniquely as a sum of 2 n2^n terms

η= Iη Idx I, \eta = \sum_I \eta_I \wedge \mathrm{d}x^I ,

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

dx I=dx I 1dx I p \mathrm{d}x^I = \mathrm{d}x^{I_1} \wedge \cdots \wedge \mathrm{d}x^{I_p}

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

Twisted and vector-valued forms

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

Of particular importance are LL-valued forms when LL is a line bundle; these are also called LL-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 LL as a basis. Therefore, they can seem sneaky and confusing sometimes when you realise that they do not behave in the same way!

Let Ψ\Psi be the pseudoscalar bundle, obtained as the bundle associated to the frame bundle by the character ρ(A)=sgn(det(A))\rho(A)=sgn(det(A)). That is, a section of Ψ\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 Ψ\Psi-twisted form is called pseudoform.

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

Given any real number ww, the bundle |Λ| w|\Lambda|^w associated to the character A|detA| wA\mapsto |det A|^{-w} is called the line bundle of ww-weighted scalars (also called w w -densities); a form twisted by this line bundle is a ww-weighted form. Note that |Λ| 0=1|\Lambda|^0=1 is the trivial bundle, so a 00-weighted form is just an ordinary form. Also, an nn-pseudoform turns out to be equivalent to a 11-weighted scalar:

Λ nΨ|Λ| 1.\Lambda^n \otimes \Psi \simeq |\Lambda|^1.

Since ΨΨ=1\Psi\otimes\Psi=1 is the trivial bundle, we have Λ n|Λ| 1Ψ\Lambda^n \simeq |\Lambda|^1 \otimes\Psi, and thus a densitised form is equivalent to a 11-weighted pseudoform:

Λ kΛ nΛ kΨ|Λ| 1\Lambda^k \otimes \Lambda^n \simeq \Lambda^k \otimes \Psi \otimes |\Lambda|^1

The bundle Λ nΨ|Λ| 1\Lambda^n\otimes\Psi\simeq|\Lambda|^1 of nn-pseudoforms is the absolute value of the bundle of nn-forms (that is of densitised 00-forms), so we may take the absolute value of one of either and get one of the latter. Similarly, the bundle of 00-forms is the absolute value of the bundle Ψ\Psi of pseudo-00-forms; that is, the trivial bundle is the absolute value of the pseudoscalar bundle.

As smooth functors on nn-paths

One way to exhibit this statement nicely is:

A differential nn-form on XX is a smooth nn-functor P n(X)B nP_n(X) \to \mathbf{B}^n \mathbb{R} from the path n-groupoid of XX to the nn-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 σ n\sigma_n is a morphism in P n(X)P_n(X), then (unless XX is a directed space), the opposite σ n 1\sigma_n^{-1} is also in P n(X)P_n(X) and I think ω(σ n)=ω(σ n 1)\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 kk-volumes as its k-morphisms, simply because these kk-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 B n\mathbf{B}^n \mathbb{R} coming from the crossed complex

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

but the more refined crossed complex

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

where the 2\mathbb{Z}_2-factor acts by sign reversal on \mathbb{R} (one can also use U(1)U(1) instead of \mathbb{R}, this way [P n(),B nU(1)][P_n(-), \mathbf{B}^n U(1)] becomes the Deligne complex and knows not just about differential forms but about U(1)U(1) n1n-1-gerbes with connection even).

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

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

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

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

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

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

Since one can let NN increase arbitrarily in this prescription – NN \to \infty – it follows that the value of the functor on Σ\Sigma is already determined by all its values on all “infinitesimal nn-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 nn-cube” a number.

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

  • In differential geometry an “infinitesimal nn-cube” is modeled by an nn-tuple of tangent vectors and a differential form is a fiberwise linear map from the nn-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 nn-cube” has a concrete meaning on the same footing of other nn-cubes. If D nD^n denotes the abstract infinitesimal nn-cube in this context, then the mapping space X D nX^{D^n} of morphisms from D nXD^n\to X is the nn-fold tangent bundle of XX and a differential form is precisely nothing but a morphism

    ω:X D n×D n \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 nn-functoriality discussed above).

    This is described in detail in section 4 of

    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 XX and YY and a continuously differentiable map f:XYf\colon X \to Y, any differential form η\eta on YY defines a pullback form f *(η)f^*(\eta) on XX. See at pullback of differential forms.

Thus, the operation that maps XX to Ω *(X)\Omega^*(X) extends to a contravariant functor Ω *\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 ff to do so; see the discussion of integration of pp-pseudoforms at integration of differential forms for an example.

Integration of forms

Let XX be an nn-dimensional manifold, and let ω\omega be an nn-pseudoform on XX. At least when XX is paracompact and Hausdorff, we may turn ω\omega into a measure on XX and thereby find its integral. Conversely, any absolutely continuous Radon measure on XX arises in this way from a unique nn-pseudoform ω\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 XX equipped with some appropriate structure. In particular, if XX itself is equipped with an orientation, then nn-pseudoforms on XX are the same as (untwisted) nn-forms, and so we can integrate those on XX. 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 MM is the space of functions on the odd tangent bundle ΠTM\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 MM 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

dC Ω 2(X)dC Ω 1(X)dC Ω 0(X)d0; \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 XX.

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 XX, giving us a homology theory. Combining these, we have Stokes's theorem

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

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



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

Textbook accounts:

Exposition with eye towards applications in mathematical physics:

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:

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:

Last revised on April 2, 2024 at 13:43:19. See the history of this page for a list of all contributions to it.