Analysis on Manifolds III: Differential Forms
Part III of a five-part lecture series on differential forms and the generalised Stokes theorem.
This lecture includes interactive SymPy cells that verify key results symbolically. SymPy is a Python library for symbolic mathematics. Loading it fetches a WebAssembly Python runtime (approx. 15 MB, cached after first load). You can also load it on demand from any code cell below.
0. From pointwise algebra to smooth fields
Part II built the algebra of a single vector space: the dual , the alternating tensors , the wedge product, the determinant, the interior product, the Hodge star, and the pullback under a linear map. Every construction lived at one point.
Part III glues those constructions together. A smooth manifold has a tangent space at every point , each of which is a real vector space of dimension . At every point we have the full machinery of Part II applied to : there is a at every point. A differential -form on assigns an element of to every point, smoothly in .
The real content of Part III is the operation that was unavailable in Part II: the exterior derivative . It is not a pointwise operation. It differentiates the coefficients of a form with respect to the manifold's smooth structure, producing a new form one degree higher. The identity , the classical identities and , and the generalised Stokes theorem of Part V all originate here.
This post: smooth manifolds (Section 1), smooth maps (Section 2), tangent vectors with two equivalent definitions (Section 3), cotangent vectors and the differential of a function (Section 4), differential forms (Section 5), the exterior derivative (Section 6), and pullback along smooth maps (Section 7). Section 8 recovers as special cases of in . Section 9 previews closed and exact forms, which open the door to de Rham cohomology. We follow Lee [1], Tu [2], and Spivak [3]; the purely algebraic perspective from Part II remains in force.
1. Smooth manifolds
A smooth manifold is a topological space that locally looks like , with the looking-like arranged so that calculus makes sense independent of the local identification.
1.1. Charts and atlases
Let be a topological space. A chart on is a pair where is open and is a homeomorphism onto an open subset of . The integer is the dimension of the chart. The coordinate functions (where is the -th standard projection) are the local coordinates of .
Think of a chart as an identification of a patch of with a patch of , so that functions on can be written as functions of real variables.
A smooth atlas on an -dimensional topological manifold is a collection of charts such that:
- (the charts cover ).
- Each is a homeomorphism.
- For every pair , the transition map
is smooth (infinitely differentiable in the sense of Part I).
The transition map compares the two coordinate descriptions of the overlap . Smoothness of all transitions is what makes "smooth on " an invariant notion.
A smooth -manifold is a second-countable Hausdorff topological space together with a maximal smooth atlas. Two smooth atlases are equivalent if their union is again a smooth atlas; the maximal atlas is the union of all atlases in an equivalence class.
The second-countable and Hausdorff hypotheses are technical: they rule out pathologies (non-paracompact spaces, the line with two origins) and are satisfied by every example a reader is likely to encounter. We will not use them again until Part IV.
with the atlas consisting of the single chart is a smooth -manifold. Every open subset inherits a smooth structure from the inclusion.
The unit sphere is a smooth -manifold. An atlas is given by stereographic projection from the north pole and the south pole :
The charts and cover . Their transition map on the overlap is the inversion on , which is smooth.
The -torus is a smooth -manifold whose charts are small patches of mapped to the quotient. On the overlaps, transition maps are translations by elements of , which are smooth.
If is smooth and has rank at every point of , then is a smooth -manifold. The implicit function theorem of Part I provides local charts: at each , some submatrix of is invertible, so is locally the graph of a smooth function of variables. The sphere with is the prototype.
The general linear group is an open subset of , hence a smooth -manifold. The special linear group is a smooth -manifold by the level-set construction above. Every Lie group (a group with a smooth manifold structure making multiplication and inversion smooth) is a manifold of this kind.
Identify antipodal points of , or equivalently take the space of lines through the origin in . The result is a smooth -manifold, with charts provided by the affine patches and coordinates . Transition maps are rational functions of the homogeneous coordinates, smooth where defined.
The Möbius band is the quotient of by the -action . The quotient is a smooth 2-manifold (charts are open strips in of length less than , diffeomorphically projected). It is a model for a nontrivial line bundle over and a standard example of a non-orientable surface.
The following spaces are not smooth manifolds, and it is worth knowing why:
- Figure of eight (two circles meeting at a point in ): at the intersection, no neighbourhood is homeomorphic to an open interval; the local topology is a cross, not a line.
- Double cone at the apex: the apex has no neighbourhood diffeomorphic to .
- The square as a subset of : the boundary corners have no smooth chart, although each face locally looks like . The square is a topological manifold with boundary, but it is not a smooth one; smooth structures on surfaces with corners require additional machinery.
- Fractal objects (Koch snowflake, Alexander's horned sphere): homeomorphic to a manifold but not diffeomorphic to any standard one; the differential structure is obstructed.
The common failure is that smoothness of transition maps cannot be achieved at the singular point.
1.2. Smooth functions on a manifold
A function is smooth if for every chart in the maximal atlas, the composite is smooth in the Part I sense. The set of smooth real-valued functions on is denoted .
is smooth if and only if for some atlas of , every coordinate description is smooth.
Proof
If is smooth for every chart in the maximal atlas, it is smooth for every chart in any sub-atlas. Conversely, if is smooth on for some atlas and is any chart in the maximal atlas, then on ,
The transition map is smooth by the atlas compatibility condition ([atlas](3)), and composition of smooth maps is smooth (chain rule, Part I), so is smooth. Coverage: every point of lies in some .
The proposition says a local-coordinate check on any one atlas is enough. This is the usual way to verify smoothness in practice.
On with coordinates inherited from the ambient space, the height function is smooth. In stereographic coordinates from the north pole, , which is smooth on .
1.3. Partitions of unity
Smooth functions on a manifold are abundant enough to glue local constructions into global ones. The tool is the partition of unity, a standard device which we record here because Part IV will use it to integrate forms.
There is a smooth function with values in such that for and .
Proof
Start with the non-analytic smooth function
All derivatives of vanish at (a standard induction), so . Define
The denominator is strictly positive for every (at least one of , is positive), so ; for and for . Set . Then where and where . Smoothness on is clear; at the function is constant on an open ball, hence smooth there too.
For any point and any open there exists with values in such that on a neighbourhood of and .
Proof
Pick a chart with and . The image contains an open ball for some . Define on and extend by zero outside. Smoothness across the boundary follows from outside ; the set is the required neighbourhood.
Let be an open cover of . A smooth partition of unity subordinate to is a family with:
- and for every .
- Local finiteness: every has a neighbourhood meeting for only finitely many .
- for every .
Every open cover of a smooth manifold admits a smooth partition of unity subordinate to it.
The proof uses second-countability to extract a countable locally finite refinement of the cover, then applies [bump-at-point] to each element and renormalises. We refer to Lee [1], Chapter 2 for the full argument.
They let us globalise: any smooth construction defined on chart patches (a smooth function, a Riemannian metric, a volume form, an integral) can be multiplied by and summed over to produce a global object. Without a partition of unity, gluing local coordinate computations on a manifold requires ad hoc care. We will use this extensively in Part IV to define .
2. Smooth maps between manifolds
Let be smooth manifolds. A map is smooth if for every there exist charts around and around with , such that the coordinate representative
is smooth in the Part I sense. A diffeomorphism is a smooth bijection with smooth inverse.
If and are smooth, then is smooth. Proof: at , pick charts , , around , , . Then , a composition of smooth maps between open subsets of Euclidean spaces, hence smooth.
The inclusion is smooth. In stereographic coordinates on , the ambient coordinates are rational functions in , hence smooth.
The case recovers [smooth-function]. Smooth maps are the elements of .
3. Tangent vectors
In a tangent vector at a point is just a vector: the velocity of a curve, or an arrow from the point. On a manifold there is no ambient space to draw arrows in, and we need an intrinsic replacement.
Two standard definitions exist, and they give the same vector space. We state both and prove equivalence.
3.1. Velocity of a curve
Let be a smooth manifold and . A smooth curve through is a smooth map from an open interval with . Two curves through are tangentially equivalent at if for some (equivalently, every) chart around ,
The set of equivalence classes is the tangent space .
The tangential equivalence of and at holds for one chart around if and only if it holds for every chart.
Proof
Let and be two charts around , with transition smooth at . By the chain rule (Part I),
Since is an invertible linear map (its inverse is , smooth by the same compatibility), the equality of derivatives in one chart transfers to the other chart by application of an invertible linear map, which preserves equality.
carries a unique real vector-space structure making the identification (via any chart) a linear isomorphism. In particular, .
Proof
Fix a chart around . The map
is well-defined (equivalent curves give the same derivative by definition) and a bijection onto : given , the curve has , showing surjectivity; injectivity is by definition.
Transport the vector-space structure of through this bijection. Uniqueness: [tangent-curve-chart-indep] shows any other chart gives the same structure up to an invertible linear isomorphism, so the structure does not depend on the chart chosen.
3.2. Derivations
The second definition avoids choosing curves at all.
A derivation at is a linear map satisfying the Leibniz rule
The set of derivations at , with pointwise linear-combination, is a real vector space .
Let be a chart around with coordinates . Define by
Linearity is clear; Leibniz follows from the product rule of calculus applied to . So each is a derivation at . We will see these form a basis of .
If is a derivation at and vanishes on an open neighbourhood of , then . In particular, depends only on the germ of at .
Proof
Let on an open . Choose a bump function with on a smaller neighbourhood and outside . Then globally (on , , so both sides are zero; outside , , so ). Applying and the Leibniz rule,
using and .
for every constant function and every derivation at .
Proof
By Leibniz, , so . By linearity, .
3.3. The two definitions agree
The map
is a well-defined linear isomorphism. Under , the basis vector (the equivalence class of a curve with ) corresponds to the partial-derivative derivation of the previous example.
Proof
Well-defined. If , then in any chart . For any smooth , by the chain rule,
so the right-hand side depends only on the equivalence class. Linearity is clear; Leibniz follows from the product rule.
Injectivity. Suppose for every . Take , the -th coordinate function of a chart around multiplied by the bump function of [bump-at-point] ( near ); this is smooth on all of , and by [derivation-local] derivation values at coincide with those of locally. Then is the -th component of , which equals zero for every . So in , meaning .
Surjectivity. Let . We show acts as a linear combination of , which proves both surjectivity and the basis claim.
Fix a chart with coordinates and (translate if needed). Set . For any , write . The multivariable Taylor expansion with integral remainder gives, for near ,
with smooth and . Pulling back via , there exist smooth on a neighbourhood of with and
Applying and using linearity, [derivation-constant], and the Leibniz rule,
since . Hence . Taking to be the curve gives .
The proof also identifies the coordinate basis of : corresponding to the coordinate directions.
For a chart with coordinates around , the coordinate basis of is . We often write synonymously.
4. The cotangent space
The cotangent space at is the dual vector space
An element of is a 1-form (or covector) at . It is a linear map .
For , the differential of at is the covector
Equivalently, under the kinematic interpretation, .
Reading as a smooth map, the pushforward (Section 7) takes a tangent vector to a tangent vector on . Under the canonical identification , coincides with . We use for the covector viewpoint and reserve for pushforwards to non-trivial targets.
In any chart around with coordinate functions , the differentials form the basis of dual to the coordinate basis of :
Proof
Apply to the coordinate function :
using , the -th coordinate of . By [dual-basis] of Part II, is the dual basis of .
For in a chart with coordinates ,
Proof
Expand in the dual basis : the coefficient of is .
5. Differential forms
Sections §3 and §4 produced two fibrewise assignments, one at each point of : the tangent space (where velocities live) and the cotangent space (where differentials of functions live). Each of these is an -dimensional real vector space, and every construction of Part II (wedge, alternator, determinant, Hodge star, pullback) applies to the alternating power .
Differential forms are the globalisation. A differential -form assigns to every point an element of , smoothly in . Three questions immediately arise:
- In what sense is the assignment smooth? The collection of all as varies is not obviously a smooth manifold; we need to give it a smooth structure first.
- How do we compute with forms? Any chart on gives a basis of at every point of the chart domain, so locally a form is a linear combination of basis wedges with smooth coefficient functions. This is what "smooth" will mean in practice.
- Why not just work in coordinates? Because the coefficients depend on the chart: the same form has different coefficients in different charts, and we need the intrinsic object to reason about global statements (is the form exact? does it extend to the whole manifold? how does it transform under a map?).
The right framework is vector bundles. A vector bundle organises a family of vector spaces parameterised by a manifold into a single smooth object. We record the general definition, list the three bundles that will matter (, , ), and then define a differential form as a smooth section of the last of these.
5.1. Vector bundles and sections
A smooth vector bundle of rank over is a smooth manifold together with a smooth surjection such that:
- Each fibre carries the structure of a real vector space of dimension .
- For every there is an open neighbourhood and a diffeomorphism (a local trivialisation) that preserves fibres, for each , and restricts to a linear isomorphism .
A smooth section of is a smooth map with . The space of smooth sections is , a -module under pointwise operations.
- The tangent bundle is a rank- vector bundle. A section is a vector field; the space of smooth vector fields is . Local trivialisations come from coordinate bases .
- The cotangent bundle is a rank- vector bundle. A section is a 1-form on ; . Local trivialisations come from coordinate covectors .
- The bundle of alternating -forms is a rank- vector bundle, with local trivialisations given by the Part II basis in any chart. A section is a differential -form.
A bundle like is built as a set from the disjoint-union data. Its smooth-manifold structure is uniquely determined by the requirement that the local trivialisations be diffeomorphisms. We record this as a background fact (the "vector bundle chart lemma"); proof in Lee [1], Chapter 10.
5.2. Differential forms in coordinates
A differential -form on is a smooth section . We write for the space of smooth -forms.
Equivalently, is an assignment that is smooth in the sense that for every chart with coordinates , writing
the coefficient functions are smooth.
By Part II, and for .
If the coefficients of are smooth in one atlas, they are smooth in every atlas: under a change of coordinates , the new coefficients are polynomials in the old coefficients and in the partial derivatives (smooth by atlas compatibility), so smoothness transfers. We will not verify this in detail; the analogous argument for tangent vectors was [tangent-curve-chart-indep].
With global coordinates :
- : smooth functions.
- .
- .
- . Each is an infinite-dimensional -module under pointwise addition and multiplication by smooth functions.
5.3. Pointwise operations on forms
Every algebraic operation of Part II extends pointwise to differential forms, producing smooth output from smooth input.
Let , , and (a smooth vector field).
- The wedge product defines a smooth element .
- The interior product defines a smooth element (with if ).
Both operations are -bilinear and inherit all structural identities from Part II (graded commutativity, associativity, , graded Leibniz).
Proof
In a chart with coordinates , expand , , and , with smooth coefficients. Distributivity of and (Part II) gives
Products and sums of smooth functions are smooth, and is a polynomial in the (specifically, a sum of signed terms), so the coefficients of both output forms are smooth in the chart. The structural identities hold pointwise by Part II and are therefore inherited.
A -form and vector fields produce a smooth function . This pairing is -multilinear, which is a stronger property than -multilinearity: multiplying a vector field by any smooth function scales the output by that same function at each point. A theorem (not proved here; see Lee [1], Lemma 12.24) gives the converse: any -multilinear, alternating map arises from a unique -form. This is the tensor characterisation of forms.
6. The exterior derivative
This is the operation of Part III. It has no pointwise analogue: it differentiates form coefficients with respect to the manifold's smooth structure and raises the degree by one.
6.1. Coordinate definition
In a chart with coordinates , for a -form (sum over increasing multi-indices, ), define
For this is the differential of a function ([df-expansion]).
This is a formula, not an intrinsic definition. The intrinsic content is the characterisation theorem below.
6.2. Axiomatic characterisation
There exists a unique family of -linear maps (one for each ) with the following properties:
- Agreement on functions. For , is the differential from [differential-function].
- Graded Leibniz rule. For and ,
- . The composition is zero.
In local coordinates, this unique coincides with the formula [eq:d-local-formula].
Proof
Existence. In a chart, define by the coordinate formula [eq:d-local-formula]. We verify the three properties, then show the definition is chart-independent so that the local piece together into a global operator.
Property (1) is immediate for : the formula reduces to , which is in coordinates by [df-expansion].
Property (2) on decomposables: take and , so . Then
using the product rule for functions. The first term is . The second is , where the sign comes from moving past (a 1-form past a -form, sign by graded commutativity). Linearity extends the identity to arbitrary forms.
Property (3) on :
The double-sum is symmetric in (by equality of mixed partials), while is antisymmetric. The sum therefore vanishes. Linearity extends to all of .
Chart independence. Suppose and are the coordinate-formula operators in two different charts around the same point, both acting on forms defined on the overlap . Both satisfy (1)-(3) above on (each property is verified purely locally by the proofs just given). The uniqueness argument in the next paragraph runs identically on and forces on the overlap. Hence the chart definitions piece together into a well-defined global operator.
Uniqueness. Suppose both satisfy (1)-(3). We show on every , locally.
In a chart, any is a finite -linear combination of ; the coefficients are smooth functions . By -linearity, it suffices to check . By the graded Leibniz rule (property 2),
On functions, by property (1). It remains to show . Iterating the Leibniz rule on and using by property (3), every term in the expansion is zero. Therefore , and hence on decomposables, hence everywhere.
6.3. Coordinate-free formula
For , define by
A short computation shows satisfies the Leibniz rule on products, so it is a derivation; by the derivation-tangent-vector correspondence ([tangent-equivalence]) it extends to a smooth vector field, the Lie bracket of and . In coordinates, . Coordinate vector fields commute: .
Let and . Then
where is the Lie bracket of vector fields, the vector field acting on functions by .
Proof
Denote by the right-hand side, so we must show as a -form.
Step 1: is -multilinear and alternating in .
-multilinearity is clear from the definition since , the bracket, and are each -multilinear. For alternation, swap and (). Each single-term index picks up a sign from the two swaps inside , which maintains alternation. Pairing the two terms and and using alternation of in its remaining slots produces the required overall sign flip. The bracket sum behaves analogously: a pair swaps to , and , so the sign is correct; the other pairs pick up signs from 's alternation.
Step 2: is -linear in the first slot.
Replace by for . Compute the change in each piece of .
Single-term sum. For ,
For , using (-linearity in the first slot of at the level of a single point, which holds pointwise as is a -multilinear map eating plain tangent vectors):
The contribution to in excess of from the single-term sum is
Bracket sum. Only pairs with contribute a change. Use (a quick computation from the bracket's definition as a commutator of derivations). So for ,
The excess over from the bracket sum is
Hence , giving .
By Step 1 alternation, -linearity in the first slot extends to -linearity in every slot.
Step 3: is a -form.
By Steps 1-2, is a -multilinear alternating map . By the tensor characterisation (remark at end of §5.3), there is a unique with pointwise. We abuse notation and write for this form.
Step 4: in coordinates.
Both sides are forms (Step 3 and [d-characterisation]), so it suffices to check they agree when evaluated on coordinate vector fields in any chart. Coordinate vector fields commute, , so the bracket sum in vanishes. We must show
By -linearity reduce to for an increasing multi-index and . By Eq. [eq:d-local-formula], , and evaluation of a -fold wedge on coordinate vector fields gives the determinant
where the determinant is of the matrix whose first row is and whose subsequent rows are (from Part II, [eq:wedge-as-det-formula]). On the right-hand side, expanding and applying gives ; the second term vanishes (the Kronecker matrix has constant entries), leaving only the first. Cofactor expansion along the first row of the matrix in the left-hand side exactly matches the signed sum on the right, with the coming from the position of in that expansion. Both sides therefore equal the same determinant.
Hence and agree on coordinate vector fields in every chart. By Step 3 both are forms, and two forms that agree on coordinate vector fields in every chart agree everywhere.
This is the Cartan formula in the case: . The bracket term is essential: without it, the right-hand side would not be -linear in (smooth functions would leak out of derivatives).
6.4. Worked examples
Let . Then
The coefficient is the scalar curl of in the plane; closedness of is the classical condition that it be locally exact.
The 1-form on is closed: a direct computation of gives . It is not exact, because its integral around the unit circle (oriented counterclockwise) equals , while the integral of an exact form around a closed loop must vanish by the fundamental theorem for line integrals (Part IV). This is the standard obstruction that makes nontrivial; the form is (up to scale) the volume form of the angular coordinate, where is only locally defined.
On with polar coordinates , the area form transforms as
The Jacobian factor appears as the coefficient; no absolute value is needed because orientation is tracked by the wedge.
6.5. The Lie derivative and Cartan's magic formula
The exterior derivative, the interior product, and a third operation (the Lie derivative) fit into a single identity.
For and , the Lie derivative measures the infinitesimal change of along the flow of :
where is the flow of at time (a one-parameter family of local diffeomorphisms of ).
For every and ,
Equivalently, on .
Proof
Write . Both and are -linear maps for each (preserving degree).
Step 1: Both operators are degree-zero derivations of the wedge product.
For on , apply the graded Leibniz rules for ([d-characterisation]) and ([iota-properties]):
Add the two. The terms with have coefficients , and the terms cancel likewise. The remaining terms give . So is a degree-zero derivation.
For : the flow is a one-parameter family of local diffeomorphisms, and by product-compatibility of pullback ([pullback-properties-smooth](2)). Differentiating in at and using the ordinary Leibniz rule gives .
Step 2: Both operators agree on .
For , (by convention, since ), so . On the other hand,
by the definition of as the velocity of its flow. Hence .
Step 3: Both operators commute with .
For : , and , using . So .
For : by [pullback-properties-smooth](5) applied to , . Differentiating at , .
Step 4: Conclusion.
Let . By Steps 1, is a degree-zero derivation of the wedge algebra. By Step 2, on . By Step 3, commutes with .
Any -form is locally a finite -linear combination of in a chart. Each is of the 0-form , on which vanishes, so . By the derivation property on wedges, for every multi-index. Since vanishes on smooth functions and on basis wedges, and is a derivation on their products, . By linearity, on every locally, hence globally.
The exterior derivative is -linear but not -linear: , not . This is a feature, not a defect: a -linear map is a tensor (pointwise operation), whereas genuinely sees the smooth structure of (partial derivatives of coefficient functions). The interior product , by contrast, is -linear in and -linear in , making it a genuinely pointwise operation.
7. Pullback of forms along smooth maps
Under a smooth , functions pull back (), and algebraically, forms pull back pointwise through the transpose of .
Let be smooth and . The pullback is
where is the differential of at (the pushforward on tangent vectors, defined by ).
For smooth , , forms on , and :
- and .
- .
- and .
- is smooth whenever is, so pullback is a map .
- Compatibility with . .
Proof
Properties (1)-(3) follow from the algebraic pullback in Part II ([pullback-properties]) applied pointwise with .
For (4), expand in coordinates. Let be a chart around with coordinates , and a chart around with coordinates . If , then by (1) and (2),
For a coordinate covector, (a 1-form on ), which is smooth by the chain rule applied to the smooth function . The coefficients are smooth by [smooth-well-defined] applied to a composition of smooth maps.
For (5), we first check agreement on 0-forms: for and ,
so . The two operators and both map ; they agree on 0-forms, and both satisfy the graded Leibniz rule by (2) and the Leibniz rule for . In a chart around with coordinates , every form is locally a sum of decomposables . Each is already exact, so
By Leibniz applied iteratively to , both operators agree on every decomposable, hence by linearity on all of .
Compatibility with (property 5) is the naturality of the exterior derivative. It is what makes "pull back then differentiate" the same as "differentiate then pull back," and it underlies the change-of-variables formula for integration in Part IV.
8. are the exterior derivative on
On with the standard inner product and orientation, the Hodge star of Part II (Section 10) identifies and , and the musical isomorphisms identify as vector fields. Under these identifications, becomes the three classical operators.
On , let and correspond to the 1-form (via the flat isomorphism) and to the 2-form (via Hodge star of ). Then:
In particular, and are instances of .
Proof
. By [df-expansion], , which is by definition of gradient.
. Apply [d-local] to :
Expanding each differential and using etc.,
The triple is exactly .
. With ,
Cyclic rearrangement gives (two transpositions) and similarly for the third. Summing,
Consequences. Applying to and using the first two equalities of [eq:grad-curl-div-eq]: , so . Applying to and using the last two equalities of [eq:grad-curl-div-eq]: , so .
This is the promise made at the start of the series made good. Three distinct-looking operators are the same operator seen through the isomorphisms between spaces on , and the two identities "curl of gradient is zero" and "divergence of curl is zero" are the single statement .
9. Orientations on a manifold
Before integration (Part IV) we need a consistent notion of "positive orientation" across charts.
Two ordered bases and of a real vector space are consistent if the change-of-basis matrix has positive determinant. This is an equivalence relation with exactly two classes; a choice of class is an orientation on . Equivalently, under the Part II correspondence, an orientation on is a connected component of .
A smooth -manifold is orientable if it admits an atlas such that every transition map has positive Jacobian determinant wherever defined. A choice of such an atlas (maximal among compatible atlases) is an orientation on , and equipped with an orientation is oriented.
Equivalently, is orientable iff there exists a nowhere-vanishing -form ; two such forms give the same orientation iff they differ by a strictly positive smooth function.
- are orientable. Explicit top forms: on ; restriction of to .
- The Möbius band and for even are not orientable: no global nowhere-vanishing top form exists. The cyclic quotient in the Möbius band forces any candidate form to reverse sign after one loop.
- Every Lie group is orientable (use left-invariant top forms).
The bundle is a two-sheeted covering of . Orientability of is the condition that this covering is trivial; a choice of trivialisation selects one sheet and hence an orientation. Non-orientable manifolds have a connected two-sheeted cover (the orientation double cover), which is always orientable. The Möbius band's orientation double cover is the cylinder .
10. Closed and exact forms
A form is closed if and exact if for some .
By every exact form is closed. The converse can fail, and the failure is a genuine invariant of the manifold.
10.1. The de Rham cohomology
The -th de Rham cohomology group of is the quotient vector space
Two closed forms represent the same class iff they differ by an exact form.
The dimension is the -th Betti number. For compact manifolds (or, more generally, manifolds of finite type) the are finite.
We saw is closed but not exact (§6.4). Its cohomology class generates ; two closed 1-forms on the punctured plane represent the same class iff their line integrals around the unit circle agree. The obstruction to exactness is a period (the integral around a loop), and the period is a continuous function of the cohomology class.
10.2. Why it matters
The quotient is built from and , which are analytic data (smooth functions, partial derivatives). A priori this quotient could depend sensitively on the smooth structure. The surprise is that it does not.
For a smooth manifold , there is a natural isomorphism
between de Rham cohomology and the real singular cohomology of the underlying topological space. In particular, is a topological invariant: it depends only on the homotopy type of , not on the smooth structure.
We state the theorem without proof; see Bott-Tu [4], Chapter I. The content for this series is the following list of pay-offs.
-
Differential equations encode topology. Closed forms are solutions to a differential equation (). Exact forms are the trivially constructible solutions (). The quotient counts, in a precise way, how many non-trivial solutions there are. The dimensions are combinatorial invariants (they equal the ranks of the simplicial homology groups for a triangulation), so a purely analytic question about forms has a purely combinatorial answer.
-
Obstruction to potentials. In physics, a closed but non-exact 1-form is a conservative-looking force field (, locally looks like a gradient) that has no global potential. The punctured-plane example is the archetype: the angular 1-form integrates to the winding number, which cannot be written as the gradient of a single-valued function. On a contractible domain every closed form is exact (Poincaré lemma, below); on a non-contractible one the failure is measured by .
-
Integration detects cohomology. Stokes's theorem (Part V) implies that ; if is closed, the right-hand side vanishes, so closed forms integrate to zero against any boundary. The integrals of closed forms over cycles (closed submanifolds without boundary) are invariants that factor through the de Rham class, giving the pairing of periods. This is the bridge between differential calculus and algebraic topology.
-
Obstructions are additive. is a graded ring under the wedge product (); the ring structure encodes how -dimensional obstructions interact with -dimensional ones. For the -torus, recovers the Part II algebra at the level of cohomology.
-
Poincaré duality and Hodge theory. On a compact oriented -manifold, via the pairing . With a Riemannian metric, each de Rham class has a unique harmonic representative (a solution of where ); Hodge theory makes the topological invariants equal to the dimensions of kernels of elliptic PDEs. This is the point where forms meet PDE.
-
A research-grade tool. Chern-Weil theory, characteristic classes, index theorems (Atiyah-Singer), gauge theory, string theory, geometric topology: all are built on top of de Rham cohomology. A working knowledge of is the ticket into modern differential geometry.
10.3. The Poincaré lemma
If is star-shaped with respect to a point (meaning the line segment from to any lies in ), then every closed form on is exact. More generally, every contractible manifold has for and .
Proof
Take after translation; is star-shaped at the origin.
The homotopy operator. Define for by
Star-shapedness guarantees for every , so the integrand is well-defined; smoothness of and smooth dependence on parameters make . For , set on .
Claim: for every , .
Write in multi-index notation with . Both sides are -linear in , so reduce to .
Let and consider the map , . On with coordinates , decompose the pullback: every form on splits as
with not containing . Pulling back via ,
Expanding the wedge and collecting -containing terms, the -component of is
and the -component is .
Integrating over (fibre integration) gives :
Set . Compute using Leibniz; each term produces , where .
Similarly, , and
After cancellation of the mixed terms against the contributions in , the sum reduces to
The integrand of the first integral is (from the chain rule for and the product rule). So the combined expression is
This establishes .
Conclusion (star-shaped case). Let be closed, . By the identity above, , so is exact with primitive .
Contractible case. A contractible manifold has a smooth homotopy from a constant map to . The same fibre-integration construction applied to gives a chain homotopy between and the constant-map pullback (which is zero on for ), so closed forms are exact on . follows from the fact that a function with is locally constant, and connected.
The practical consequence: locally, on any chart, every closed form is exact. De Rham cohomology is the obstruction to patching local primitives into a global one.
11. Looking ahead
Part IV integrates -forms over oriented -dimensional regions inside . The pullback compatibility ([pullback-properties-smooth](5)) will give the change-of-variables theorem for free. Part V combines all of it into the generalised Stokes theorem , recovering the fundamental theorem of calculus ( on ), Green's theorem ( on ), the divergence theorem ( on ), and the classical Stokes theorem for surfaces as special cases.
References
- J.M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Chapters 1-3, 11-14. DOI
- L.W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. Chapters 1-4. DOI
- M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd ed., Publish or Perish, 1999.
- R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. Chapter 1.
- M. Spivak, Calculus on Manifolds, W.A. Benjamin, 1965. DOI