nLab absolute differential form

Absolute differential forms

Absolute differential forms


It's well known that one can integrate a differential form on an oriented submanifold. Less well known (but also true), one can integrate a differential pseudoform on an pseudoriented (transversely oriented) submanifold. But in classical differential geometry, one also sees forms that can be integrated on unoriented submanifolds.

I call these absolute forms. The term ‘absolute’ suggests a lack of additional required structure, in this case some sort of orientation on the domain of integration. It also suggests absolute value, since many of the examples from classical differential geometry involve absolute values. Indeed, we can define the absolute value of a form or a pseudoform to be an absolute form, although not every absolute form arises in this way.

The main theorem of absolute forms is that, if ω\omega is a (pseudo)-pp-form and RR is a (pseudo)-oriented pp-dimensional submanifold, then

| Rω| |R||ω|, {\left| \int_R \omega \right|} \leq \int_{|R|} {|\omega|} ,

where |ω|{|\omega|} is an absolute pp-form (the absolute value of ω\omega), |R||{R}| is simply RR with its (pseudo)-orientation ignored, and the absolute value on the left is the ordinary absolute value of scalars. This theorem also applies if we start with an absolute pp-form ω\omega, (although in that case RR starts out unoriented and so is the same as |R|{|R|}). If RR is a de Rham chain (a formal linear combination of appropriately oriented submanifolds), we also take absolute values of the formal coefficients in |R|{|R|}. (This operation does not respect the usual notion of equality of chains, but the theorem is true all the same.)


Let XX be a differentiable manifold (or similar sort of space), and let pp be a natural number (typically 0pn0 \leq p \leq n, where nn is the dimension of XX). Recall that an (exterior differential) pp-form ω\omega on XX is a function that assigns a real number (or whatever is the relevant sort of scalar) ω c(v 1,,v p)\omega_c(v_1,\ldots,v_p) to a point cc in XX and a pp-tuple (v 1,,v p)(v_1,\ldots,v_p) of tangent vectors at cc, multilinearly and alternating in the v iv_i. Similarly, a pp-pseudoform ω\omega on XX is a function that assigns a scalar ω c o(v 1,,v p)\omega_c^o(v_1,\ldots,v_p) to a point cc in XX, a local orientation oo at cc, and a pp-tuple (v 1,,v p)(v_1,\ldots,v_p) of tangent vectors at cc, multilinearly and alternating in the v iv_i and reversing sign under a reversal of oo.


An absolute pp-form ω\omega on XX is a function that assigns a scalar ω c(v 1,,v p)\omega_c(v_1,\ldots,v_p) to a point cc in XX and a pp-tuple (v 1,,v p)(v_1,\ldots,v_p) of tangent vectors at cc and that satisfies the following conditions:

  1. Fixing cc, ω c()\omega_c({-}) shall be uniformly continuous.

  2. The pp-tuple (v 1,,v p)(v_1,\ldots,v_p) shall be linearly independent if ω c(v 1,,v p)0\omega_c(v_1,\ldots,v_p) \ne 0. Thus, although ω c\omega_c is not linear, we may still call it alternating; however (as a consequence of 3), it is actually symmetric.

  3. Fix a pp-dimensional subspace SS of the tangent space at cc and an orientation oo of SS. Now given a linearly independent pp-tuple (v 1,,v p)(v_1,\ldots,v_p) from SS (that is a basis of SS), let ω c(v 1,,v p) S o\omega_c(v_1,\ldots,v_p)_S^o be ±ω c(v 1,,v p)\pm\omega_c(v_1,\ldots,v_p) according to whether the orientation of SS induced by the v iv_i matches oo, and extend this by continuity to all pp-tuples from SS (which extension must be unique and exists by 1&2). The resulting function ω c() S o\omega_c({-})_S^o shall be multilinear (and so also alternating, by 2).

The multilinearity condition here is rather weaker than for a (pseudo)-form, since it applies only within a pp-dimensional subspace. Shifting one vector even slightly outside of SS loses all connection provided by multilinearity, which is why we need a continuity condition; continuity holds for (pseudo)-forms automatically.

An absolute pp-form ω\omega is continuous if it is jointly continuous in all of its data (cc as well as the v iv_i). Since the domain of the function ω\omega is a manifold (a vector bundle over XX, although ω\omega is not a map of vector bundles), we can even discuss differentiability, smoothness, and even analyticity of ω\omega when XX has the relevant structure.

An absolute 00-form is the same thing as a 00-form. An absolute nn-form on an nn-dimensional manifold XX is essentially the same thing as an nn-pseudoform; with the notation from condition 3, the only possibility for SS is the entire tangent space T cXT_c{X}, and we have

ω˜ c o(v 1,,v n)=ω c(v 1,,v n) T cX o \tilde\omega_c^o(v_1,\ldots,v_n) = \omega_c(v_1,\ldots,v_n)_{T_c{X}}^o

to relate the nn-pseudoform ω˜\tilde{\omega} to the absolute nn-form ω\omega. Finally, the only absolute pp-form for p>np \gt n is 00.

At a point cc, an absolute pp-form ω\omega is:

  • indefinite if ω c(v 1,,v p)>0\omega_c(v_1,\ldots,v_p) \gt 0 for some (necessarily linearly independent) pp-tuple of vectors and ω c(v 1,,v p)<0\omega_c(v_1,\ldots,v_p) \lt 0 for some pp-tuple,

  • semidefinite if not indefinite,

  • definite (and hence semidefinite) if ω c(v 1,,v p)0\omega_c(v_1,\ldots,v_p) \ne 0 for every independent pp-tuple of vectors at cc,

  • positive (and hence semidefinite) if ω c(v 1,,v p)0\omega_c(v_1,\ldots,v_p) \geq 0 for every pp-tuple of vectors (it is enough when they are independent),

  • negative (and hence semidefinite) if ω c(v 1,,v p)0\omega_c(v_1,\ldots,v_p) \leq 0 for every (independent) pp-tuple of vectors.

All these are at a point cc; ω\omega satisfies the condition tout court if it holds for all cc.

Given an absolute pp-form ω\omega, its absolute value |ω|{|\omega|} is a positive semidefinite absolute pp-form:

|ω| c(v 1,,v p)|ω c(v 1,,v p)|. {|\omega|}_c(v_1,\ldots,v_p) \coloneqq {|\omega_c(v_1,\ldots,v_p)|} .

If we start with a pp-form ω\omega, then the same definition defines a positive absolute pp-form |ω|{|\omega|}. If we start with a pp-pseudoform ω\omega, then essentially the same definition still works; we use either orientation to evaluate ω\omega with the same result. Note that |ω|{|\omega|} is continuous if ω\omega is. However, we may not conclude that |ω|{|\omega|} is differentiable just because ω\omega is differentiable (or even analytic). On the other hand, |ω|{|\omega|} inherits differentiability properties from ω\omega wherever ω0\omega \ne 0. (Even then, however, we cannot inherit analyticity, except in 11 dimension.)

Given two absolute pp-forms ω\omega and η\eta, their sum ω+η\omega + \eta is an absolute pp-form:

(ω+η) c(v 1,,v p)ω c(v 1,,v p)+η c(v 1,,v p). (\omega + \eta)_c(v_1,\ldots,v_p) \coloneqq \omega_c(v_1,\ldots,v_p) + \eta_c(v_1,\ldots,v_p) .

Given an absolute pp-form ω\omega and a scalar field ff, their product fωf \omega is an absolute pp-form:

(fω) c(v 1,,v p)f(c)ω c(v 1,,v p). (f \omega)_c(v_1,\ldots,v_p) \coloneqq f(c) \omega_c(v_1,\ldots,v_p) .

In this way, the space of absolute pp-forms is a module over the algebra of scalar fields and the space of sections of a vector bundle. For now, we decline to define products of absolute forms of aribtrary rank.

Given an absolute pp-form ω\omega on XX, a manifold UU, and a continuously differentiable map R:UXR\colon U \to X, the pullback R *ωR^*\omega is an absolute pp-form on UU:

(R *ω) c(v 1,,v p)ω R(c)(R *v 1,,R *v p). (R^*\omega)_c(v_1,\ldots,v_p) \coloneqq \omega_{R(c)}(R_*v_1,\ldots,R_*v_p) .

Here, R *v iR_*v_i is the pushforward of v iv_i under RR. Note that R *ωR^*\omega is continuous if ω\omega is; we can also pull back differentiability and analyticity properties that ω\omega and RR both have.

Given a continuous absolute pp-form ω\omega on XX, a pp-dimensional manifold UU, and a continuously differentiable map R:UXR\colon U \to X, the integral Rω\int_R \omega is a scalar:

Rω UR *ω. \int_R \omega \coloneqq \int_U R^*\omega .

On the right-hand side, R *ωR^*\omega is a continuous absolute pp-form on UU, but since UU is pp-dimensional, this is essentially the same as a continuous pp-pseudoform on UU, and we already know how to integrate this (see integration of differential forms).


Examples of absolute forms from classical differential geometry include:

  • Absolute 00-forms are the same as ordinary 00-forms.

  • Absolute nn-forms on an nn-dimensional manifold are the same as nn-pseudoforms (and hence the same as absolutely continuous Radon measures).

  • In complex analysis, |dz|{|\mathrm{d}z|} is an absolute 11-form sometimes used in contour integration. This literally is the absolute value of the differential of the identity map zz.

  • More generally, the arclength element ds=dx\mathrm{d}s = {\|\mathrm{d}\mathbf{x}\|} on a Riemannian manifold is an absolute 11-form. Neither ds\mathrm{d}s nor (in general) dx\mathrm{d}\mathbf{x} is actually the differential of anything, but dx\mathrm{d}\mathbf{x} is the canonical vector-valued 11-form (which, on an affine space, really is the differential of the identity map x\mathbf{x}), and we really can use the metric to take the norm of such a form to get an absolute 11-form.

  • Similarly, the surface area? element dS\mathrm{d}S on a Riemannian manifold is an absolute 22-form, and we can continue into higher dimensions (although the classical volume element dV\mathrm{d}V in 3\mathbb{R}^3 is already covered as a 33-pseudoform). In principle, we ought to be able to write down expression for dS\mathrm{d}S etc in terms of ds\mathrm{d}s, although so far the only thing that I know how to do is dS=dx×^dx/2\mathrm{d}S = {\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2, where ×^\hat\times indicates a wedge product of vector-valued forms whose vectors are multiplied by the cross product. (This can be generalized to any finite-dimensional area in any finite-dimensional Riemannian manifold; in particular, dV=|dx^dx×^dx|/6\mathrm{d}V = {|{\mathrm{d}\mathbf{x} \hat\cdot \mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}|} / 6.)

The reason that ds\mathrm{d}s determines dS\mathrm{d}S is that lengths determine areas, as through Heron's Formula. However, our usual ways of deriving one form from another (thinking of them, at a given point, as maps from lists of vectors to numbers) involve only rerranging (permuting, duplicating, and contracting) the inputs and applying operations to the outputs. But this cannot be sufficient; knowing only the lengths of two vectors vv and ww (which vectors are the inputs to dS\mathrm{d}S, and which lengths are the outputs of ds\mathrm{d}s at those inputs) does not tell you the area of the parallelogram that they span (which is the output of dS\mathrm{d}S). You also need the length of vwv - w (or of v+wv + w). So while we can express dS c\mathrm{d}S_c (dS\mathrm{d}S at a point cc) as (v,w2H(ds(v),ds(w),ds(vw))(v, w \mapsto 2 H(\mathrm{d}s(v), \mathrm{d}s(w), \mathrm{d}s(v - w)) (where HH is the Heron function, the real-valued function of three real variables that takes the lengths of the three sides of a triangle and returns its area), or equivalently as (v,w2H(v,w,vw))(v, w \mapsto 2 H({\|v\|}, {\|w\|}, {\|{v - w}\|})), we cannot express dS\mathrm{d}S with a formula anything along the lines of dx×^dx/2{\|{\mathrm{d}\mathbf{x} \hat\times \mathrm{d}\mathbf{x}}\|} / 2 using ds\mathrm{d}s directly instead of dx\mathrm{d}\mathbf{x}. (And so you may as well describe dS c\mathrm{d}S_c as (v,wv×w)(v, w \mapsto {\|{v \times w}\|}) instead.)


Near the end of a Usenet post from 2002, we see a definition of R|ω|\int_R {|\omega|} for ω\omega a (pseudo)-pp-form and RR a pp-dimensional submanifold, but without a broader context for |ω|{|\omega|} itself:

Apparently absolute pp-forms (at least if continuous) are the same as even pp-densities as defined by Gelfand; see this MathOverflow answer:

Last revised on May 15, 2023 at 06:19:29. See the history of this page for a list of all contributions to it.