# Differential form

A differential form of degree $p$, a $p$- form, on a differentiable manifold $M$ is a $p$ times covariant tensor field on $M$. It may also be interpreted as a $p$- linear (over the algebra $F( M)$ of smooth real-valued functions on $M$) mapping ${\mathcal X} ( M) ^ {p} \rightarrow F( M)$, where ${\mathcal X} ( M)$ is the $F( M)$- module of smooth vector fields on $M$. Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential $df$ of a smooth function $f$ on $M$, which is defined as follows: $( df ) ( X)$, $X \in {\mathcal X} ( M)$, is the derivative $Xf$ of $f$ in the direction of the field $X$. Riemannian metrics on a manifold $M$ serve as examples of symmetric differential forms of degree two. However, the term "differential form" is often used to denote skew-symmetric or exterior differential forms, which have the greatest number of applications.

If $( x ^ {1} \dots x ^ {n} )$ is a local system of coordinates in a domain $U \subset M$, the forms $dx ^ {1} \dots dx ^ {n}$ constitute a basis of the cotangent space $T _ {x} ( M) ^ {*}$, $x \in U$. For this reason (cf. Exterior algebra) any exterior $p$- form $\alpha$ may be written in $U$ in the form

$$\tag{1 } \alpha = \sum _ {i _ {1} \dots i _ {p} } a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } ,$$

where the $a _ {i _ {1} \dots i _ {p} }$ are functions on $U$. In particular,

$$df = \frac{\partial f }{\partial x ^ {i} } dx ^ {i} .$$

Let $E ^ {p} = E ^ {p} ( M)$ be the space of all exterior $p$- forms of class $C ^ \infty$, where $E ^ {0} ( M) = F ( M)$. The exterior multiplication $\alpha \wedge \beta$ converts $E ^ {*} ( M) = \sum _ {p = 0 } ^ {n} E ^ {p} ( M)$( where $n = \mathop{\rm dim} M$) to an associative graded algebra over $F ( M)$ which satisfies the condition of graded commutativity

$$\tag{2 } \alpha \wedge \beta = ( - 1 ) ^ {pq} \beta \wedge \alpha ,\ \ \alpha \in E ^ {p} ,\ \beta \in E ^ {q} .$$

A smooth mapping between manifolds $f : M \rightarrow N$ induces a homomorphism $f ^ { * } : E ^ {*} ( N) \rightarrow E ^ {*} ( M)$ between $\mathbf R$- algebras.

The concept of the differential of a function is generalized as follows. For any $p \geq 0$ there exists a unique $\mathbf R$- linear mapping $d : E ^ {p} \rightarrow E ^ {p+1}$( exterior differentiation), which for $p = 0$ coincides with the differential introduced above, with the following properties:

$$d ( \alpha \wedge \beta ) = d \alpha \wedge \beta + ( - 1 ) ^ {p} \alpha \wedge d \beta ,$$

$$\alpha \in E ^ {p} ,\ \beta \in E ^ {q} ,\ d ( d \alpha ) = 0 .$$

The exterior differential of a form $\alpha$ written in local coordinates (see (1)) is expressed by the formula

$$d \alpha = \sum _ {i _ {1} \dots i _ {p} } da _ {i _ {1} \dots i _ {p} } \wedge dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } .$$

Its coordinate-free notation is

$$d \alpha ( X _ {1} \dots X _ {p+1} ) =$$

$$= \ \sum _ {i = 1 } ^ { {p } + 1 } (- 1) ^ {i+1} X _ {i} \alpha ( X _ {1} \dots \widehat{X} _ {i} \dots X _ {p+1} ) +$$

$$- \sum _ {i < j } (- 1) ^ {i+j} \alpha ( [ X _ {i} , X _ {j} ] , X _ {1} \dots \widehat{X} _ {i} \dots \widehat{X} _ {j} \dots X _ {p+1} ) ,$$

where $X _ {1} \dots X _ {p+1} \in {\mathcal X} ( M)$. The Lie derivative operator $L _ {X}$, $X \in {\mathcal X} ( M)$, on differential forms is connected with the exterior differentiation operator by the relation

$$L _ {X} = d \circ \iota _ {X} + \iota _ {X} \circ d ,$$

where $\iota _ {X} : E ^ {p} \rightarrow E ^ {p-1}$ is the operator of interior multiplication by $X$:

$$( \iota _ {X} \alpha ) ( X _ {1} \dots X _ {p-1} ) = \alpha ( X , X _ {1} \dots X _ {p-1} ) ,$$

$$\alpha \in E ^ {p} ( M) ,\ X _ {1} \dots X _ {p-1} \in {\mathcal X} ( M) .$$

The complex $( E ^ {*} ( M) , d )$ is a cochain complex (the de Rham complex). The cocycles of this complex are said to be closed forms, while the coboundaries are known as exact forms. According to the de Rham theorem, the cohomology algebra

$$H ^ {*} ( M) = \sum _ {p = 0 } ^ { n } H ^ {p} ( M)$$

of the de Rham complex is isomorphic to the real cohomology algebra $H ^ {*} ( M, \mathbf R )$ of the manifold $M$. In particular, $H ^ {p} ( \mathbf R ^ {n} ) = 0$ if $p > 0$( Poincaré's lemma).

The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let $D$ be a bounded domain in $\mathbf R ^ {p}$ and let $s$ be a smooth mapping $\mathbf R ^ {p} \rightarrow M$, defined in a neighbourhood of the closure $\overline{D}\;$. If $\alpha \in E ^ {p} ( M)$, then $s {} ^ \star \alpha = a dx ^ {1} \wedge \dots \wedge dx ^ {p}$, where $a$ is a smooth function in $\overline{D}\;$. The integral of the form $\alpha$ over the surface $s$ is defined by the formula:

$$\int\limits _ { s } \alpha = \int\limits _ { D } a ( x _ {1} \dots x _ {p} ) dx ^ {1} \dots dx ^ {p} .$$

If the boundary of $D$ is piecewise smooth, the formula

$$\tag{3 } \int\limits _ { s } d \alpha = \int\limits _ {\partial s } \alpha ,\ \ \alpha \in E ^ {p-1} ( M) ,$$

is valid; here $\int _ {\partial s } \alpha$ is defined as the sum of the integrals of the form $\alpha$ over the smooth pieces of the boundary, provided with their natural parametrizations. The classical formulas of Newton–Leibniz, Green–Ostrogradski and Stokes (see also Stokes theorem) are all special cases of this formula. By virtue of formula (3) each closed $p$- form $\alpha$ defines a $p$- dimensional singular cocycle whose value on the simplex $s$ is $\int _ {s} \alpha$. This correspondence is a realization of the isomorphism given by de Rham's theorem.

Formula (3) was published in 1899 by H. Poincaré , who regarded exterior forms as integrand expressions in integral invariants. At the same time E. Cartan  gave an almost-modern definition of exterior forms and of the exterior differentiation operator (at first on Pfaffian forms), stressing the connection between his own construction and exterior algebra.

As well as the exterior scalar forms defined above, one may also study exterior differential forms with values in a vector space $V$ over $\mathbf R$. If $V$ is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space $E ( M , V )$ of forms with values in $V$. If the algebra $V$ is also associative, $E ( M , V )$ is associative as well; if $V$ is commutative, $E ( M , V )$ is graded-commutative (formula (2)); if $V$ is a Lie algebra, then $E ( M , V )$ is a graded Lie algebra. The following, even more general, concept is also often considered. Let $F$ be a smooth vector bundle with base $M$. If for each point $x \in M$ there is given a skew-symmetric $p$- linear function on $T _ {x} ( M)$ with values in the fibre $F _ {x}$ of the bundle $F$, a so-called $F$- valued $p$- form is obtained. An $F$- valued $p$- form can also be interpreted as a $p$- linear (over $F ( M)$) mapping of the module ${\mathcal X} ( M) ^ {p}$ into the module of smooth sections of $F$. The space of such forms is denoted by $E ^ {p} ( F )$. If $F$ is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on $F$, it is possible to define the de Rham complex and to generalize the de Rham theorem to this case.

Forms with values in the tangent bundle $T ( M)$ are also called vector differential forms; these forms may be identified with $p$ times covariant and one time contravariant tensor fields on $M$ which are skew-symmetric with respect to the covariant indices. Vector differential forms are used to describe the derivations of the algebra of exterior forms $E ( M)$. Vector forms (as well as their generalization — jet forms) are used in the theory of deformations of complex and other differential-geometric structures on manifolds.

Analogues of differential forms are also constructed in simplicial theory. One such construction, whose idea is due to H. Whitney , may be used to calculate the rational cohomology of a simplicial complex $K$. A piecewise-linear form (or $PL$- form) on $K$ is a compatible family of differential forms defined on the simplices of the complex $K$, with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The $PL$- forms on $K$ form a graded commutative differential algebra $E _ {PL} ^ {*} ( K)$ over $\mathbf Q$. The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra $H ^ {*} ( | K | , \mathbf Q )$, where $| K |$ is the polyhedron corresponding to the complex $K$. The algebra $E _ {PL} ^ {*} ( K)$ also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of $| K |$. In a similar manner, the algebra $E ^ {*} ( M)$ on a differentiable manifold $M$ defines the real homotopy type of $M$, .

The calculus of exterior forms on a complex analytic manifold has a number of special features . In this situation it is usual to consider the space $E ^ {p} ( M , \mathbf C )$ of complex-valued forms, or the spaces $E ^ {p} ( F )$ where $F$ is a holomorphic vector bundle on $M$. The following decomposition is valid:

$$E ^ {p} ( M , \mathbf C ) = \sum _ {r+ s = p } E ^ {r , s } ( M) ,$$

where $E ^ {r , s } ( M)$ is the space of forms of type $( r , s )$, i.e. of forms $\alpha$ which are locally representable as

$$\sum a _ {i _ {1} \dots i _ {r} , j _ {1} \dots j _ {s} } \ dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {r} } \wedge d {\overline{z}\; } {} ^ {j _ {1} } \wedge \dots \wedge d {\overline{z}\; } {} ^ {j _ {s} } ,$$

where $( z ^ {1} \dots z ^ {n} )$ is a local analytic coordinate system on $M$. Similarly,

$$E ^ {p} ( F ) = \sum _ {r + s = p } E ^ {r , s } ( F ) .$$

Further, $d = d ^ \prime + d ^ {\prime\prime}$, where

$$d ^ \prime : E ^ {r , s } ( M) \rightarrow E ^ {r + 1 , s } ( M) ,\ \ d ^ {\prime\prime} : E ^ {r , s } ( M) \rightarrow E ^ {r , s + 1 } ( M) .$$

Here ${d } ^ \prime 2 = {d } ^ {\prime\prime} 2 = 0$, so that $d ^ \prime$ and $d ^ {\prime\prime}$ define cochain complexes. The best known is the complex of the operator $d ^ {\prime\prime}$( the Dolbeault complex), the cohomology of which is denoted by $H ^ {r , s } ( M)$. $d ^ {\prime\prime}$- cocycles of type $( p , 0 )$ are holomorphic $p$- forms (cf. Holomorphic form). The following Grothendieck lemma is valid for $d ^ {\prime\prime}$: If $\alpha$ is a form of type $( r , s )$ with $s > 0$ in a neighbourhood of zero of the space $\mathbf C ^ {n}$ and $d ^ {\prime\prime} \alpha = 0$, then a smaller neighbourhood of zero contains a form $\beta$ of type $( r , s - 1 )$ such that $\alpha = d ^ {\prime\prime} \beta$. The Dolbeault complex may also be defined for $F$- valued forms where $F$ is a holomorphic vector bundle. This leads to the cohomology spaces $H ^ {r , s } ( F )$. The Grothendieck lemma implies the following isomorphism:

$$H ^ {r , s } ( F ) \cong H ^ {s} ( M , \Omega ^ {r} ( F ) ) ,$$

where $\Omega ^ {r} ( F )$ is the sheaf of germs of holomorphic $F$- valued $r$- forms (Dolbeault's theorem). In particular,

$$H ^ {r , s } ( M) \cong H ^ {s} ( M , \Omega ^ {r} ( M) ) ,$$

where $\Omega ^ {r} ( M)$ is the sheaf of germs of holomorphic $r$- forms on $M$. There exists a spectral sequence with first term $\sum _ {r , s } H ^ {r , s } ( M)$ and converging to $H ^ {*} ( M , \mathbf C )$. The Euler characteristic $\chi ( M)$ of a compact complex manifold $M$ is expressed in terms of the Dolbeault cohomology spaces by the formula

$$\chi ( M) = \sum _ {r , s } (- 1) ^ {r+s} \mathop{\rm dim} H ^ {r , s } ( M) .$$

Differential forms are an important component of the apparatus of differential geometry , . They are also systematically employed in topology, in the theory of differential equations, in mechanics, in the theory of complex manifolds, and in the theory of functions of several complex variables. Currents are a generalization of differential forms, similar to generalized functions. The algebraic analogue of the theory of differential forms (cf. Derivations, module of) makes it possible to define differential forms on algebraic varieties and analytic spaces (cf. Differential calculus (on analytic spaces)). See also de Rham cohomology; Differential on a Riemann surface; Harmonic form; Holomorphic form; Laplace operator.

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How to Cite This Entry:
Differential form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_form&oldid=51183
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article