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When the Poisson equation for differential forms has one solution ${\alpha }_0$, all solutions are its translates by the kernel of a linear map of the Hodge Laplacian, namely the harmonic differential forms. The Hodge decomposition theorem identifies this translation vector space canonically with de Rham cohomology. There is no preferred affine origin until an extra normalization is chosen; orthogonality to harmonic forms picks the Green-operator solution.

Let $S^1$ act on $M\times S^1$ by rotation of the second factor. Averaging a differential form over this action is a cochain map. Every rotation is homotopic to the identity through smooth rotations, and integrating Cartan's magic formula along them gives a cochain homotopy between the averaging map and the identity. Consequently every de Rham cohomology class has a rotation-invariant representative, and an invariant exact form has an invariant primitive obtained by averaging any primitive.

Averaging a Kähler form over translations with normalized Haar measure gives an invariant Kähler form in the same de Rham cohomology class. Positivity is preserved by averaging positive values on $(v,Jv)$, and the class is unchanged because translations are homotopic to the identity. Thus any Hodge metric on a complex torus can be replaced by an invariant one.

Closed holomorphic one-forms inject into de Rham cohomology: an exact one would have a global holomorphic primitive and hence vanish. If a holomorphic class equals an antiholomorphic class, their difference is $df$, and $\partial \bar{\partial }f=0$; compact pluriharmonic rigidity makes both forms vanish. Their direct sum gives the lower bound. The sheaf of closed holomorphic one-forms exact sequence gives $0\to H^0({\Omega }^1)\to H^1(\mathbb{C})\to H^1(\mathcal{O})$, yielding the upper bound and $h^{1,0}\le h^{0,1}$. These inequalities do not require a Kähler manifold.

A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.

Suppose a nonempty compact $(n+1)$-manifold with a volume form has $n$ pointwise independent volume-preserving vector fields that commute and are tangent to its boundary. If restriction $H_{\mathrm{dR}}^1(M)\to H_{\mathrm{dR}}^1(\partial M)$ is injective, the boundary must have at least two connected components.

Indeed $\eta ={\iota }_{X_1}\cdots {\iota }_{X_n}\omega$ is nowhere zero. Cartan's magic formula and commutation show $d\eta =0$, while boundary tangency gives $j^{\ast }\eta =0$. Injectivity in de Rham cohomology makes $\eta =df$. On a connected boundary $f$ is constant; a maximum or minimum differing from that boundary value would be an interior critical point, contradicting $df\ne 0$. Empty boundary is also impossible.

On a good cover, combine Čech degree $p$ and differential-form degree $q$ in the double complex of forms on intersections. The Čech differential and exterior derivative commute, so the total differential $\delta +(-1{)}^{p}d$ squares to zero. A partition of unity and the Poincaré lemma identify its cohomology with both constant-sheaf and de Rham cohomology. The sign convention makes Čech-de Rham curvature descent explicit.

A closed differential form satisfies $d\alpha =0$; an exact differential form has $\alpha =d\beta$. The identity $d^2=0$ makes every exact form closed. Global failure of the converse is measured by de Rham cohomology; the Poincaré lemma gives the local converse in positive degrees.

Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the Poincaré lemma supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of de Rham cohomology.

If $[\sigma ]\ne 0$ in de Rham cohomology, a twisted cotangent symplectic form has no Lagrangian submanifold whose projection is a diffeomorphism onto the entire base. Such a submanifold would be a global one-form graph, and the twisted Lagrangian graph criterion would make $\sigma$ exact.

A compact Lagrangian submanifold $L$ with vanishing first de Rham cohomology cannot be a fibre of a smooth positive-dimensional fiber bundle whose nearby fibres are also Lagrangian submanifolds. In a Weinstein neighborhood, nearby fibres are one-form graphs of closed forms. They are exact by the cohomology assumption, so meet the zero section at a critical point of their potential, contradicting disjointness of distinct fibres.

For a real or complex line bundle, its endomorphism bundle is canonically trivial by scalar multiplication. The Bianchi identity therefore reduces to $d{F}_A=0$. The difference of two connections on a vector bundle is a global endomorphism-valued one-form, which here is an ordinary scalar one-form $a$. Since scalar one-forms have vanishing wedge square, the curvature difference formula reduces to $F_{A+a}-F_A=da$. Thus the de Rham cohomology class is independent of the connection on a vector bundle.

For a smooth map $f:M\to N$ between oriented connected closed manifolds of the same dimension, a normalized volume form on the target gives $\mathrm{deg}f={\int }_M{f}^{\ast }\omega$. The formula agrees with the degree as a sum of local degrees. A change of normalized top-form adds an exact form by top-dimensional de Rham cohomology, whose pullback has zero integral by Stokes theorem.

Restriction of differential forms to the finitely many open and closed components gives an isomorphism of cochain complexes ${\Omega }^{\ast }(⨆M_j)\cong {⨁}_j{\Omega }^{\ast }(M_j)$. Piecewise forms glue because there are no overlaps, and the exterior derivative acts componentwise. Taking the quotient of closed differential forms by exact differential forms proves the displayed de Rham cohomology isomorphism. For an infinite disjoint union, forms and cohomology give products instead.

A differential form is a smooth field of alternating multilinear maps that can be integrated over an appropriately oriented manifold of matching dimension. A one-form measures a directed displacement; a two-form measures an oriented area element. The point is to express what is being accumulated along a curve or over a surface without tying the answer to one set of coordinates. This makes line integrals, surface integrals, and the integral theorems of vector calculus parts of one construction.

For a concrete one-form, take $\alpha =xdy$ in the plane. Travel from $(0,0)$ to $(1,1)$ along the bottom and then the right edge of the unit square. The horizontal leg contributes zero because $dy=0$; the vertical leg has $x=1$ and contributes ${\int }_0^{1}dy=1$. Taking the left and then the top edge instead gives zero: the vertical leg has $x=0$, and the horizontal leg again has $dy=0$. The two paths have the same endpoints but different integrals.

Now replace $\alpha$ with the exact differential $\beta =d(xy)=ydx+xdy$. Both paths give $1$, because integrating an exact differential along a curve gives the endpoint difference of its potential. Here that difference is $xy{|}_{(1,1)}-xy{|}_{(0,0)}=1$. This is the fundamental theorem of calculus applied to a function along a parametric curve, and is the basic reason conservative vector fields have path-independent work.

The exterior derivative records the local obstruction in the first example: $d(xdy)=dx\wedge dy$. The Generalized Stokes theorem turns the difference between the two path integrals into the integral of this two-form over the square, whose oriented area is $1$. Interior edges cancel when adjoining pieces are added; only the outer boundary survives. That cancellation connects a local calculation with a global measurement.

More precisely, a degree-$k$ differential form assigns to each point $p$ a multilinear scalar-valued function of $k$ tangent vectors at $p$. Interchanging two arguments changes its sign. A one-form is therefore a covector field, while a two-form vanishes on a pair of parallel vectors. Degree zero simply means a smooth function. The space ${\Omega }^k(M)$ is a vector space; the wedge product of differential forms combines degrees, and the exterior derivative raises degree by one.

The usefulness goes beyond bookkeeping. Pullbacks of a differential form supply the right transformation law automatically under a coordinate change. The Generalized Stokes theorem unifies the fundamental theorem of calculus, Green theorem, Stokes theorem, and divergence theorem. Meanwhile, the distinction between closed differential forms and exact differential forms detects holes through de Rham cohomology. Reyer Sjamaar develops these connections in sections 5.4 and 9.3 of his university notes. ^[ref].

A closed differential form of degree one on the two-sphere has primitives on its north-punctured and south-punctured charts by the Poincaré lemma. The overlap is connected, so their difference is constant. Adjust one constant and glue the primitives. Thus every closed one-form is an exact differential form, proving the first de Rham cohomology vanishes.

On a compact Kähler manifold, the difference of cohomologous Kähler forms is a real, de Rham cohomology exact $(1,1)$-form. The ddbar lemma gives $\tilde{\omega }-\omega =\partial \bar{\partial }h$. Reality implies $\partial \bar{\partial }\bar{h}=-(\tilde{\omega }-\omega )$, so $f=(h-\bar{h})/(2i)$ is real and has the displayed potential equation. The difference of two real potentials is a pluriharmonic function, hence constant on each connected component. On a connected manifold the only ambiguity is one additive constant.

On a closed oriented Riemannian manifold, a smooth isometric action of a connected Lie group fixes every harmonic differential form. Each group element is connected to the identity through a smooth path, so its action is smoothly homotopic to the identity and fixes de Rham cohomology. Isometries preserve the Hodge Laplacian; uniqueness of the harmonic representative, from the Hodge decomposition theorem, then makes each form itself fixed.

A harmonic one-form is a differential one-form $\alpha$ satisfying $\Delta \alpha =0$ for the Hodge Laplacian. On a closed manifold, the Hodge decomposition theorem identifies these forms with degree-one de Rham cohomology. The Bochner-Weitzenbock formula for one-forms implies that harmonic one-forms are parallel under nonnegative Ricci curvature.

A Hodge metric is a Kähler metric whose Kähler form represents an integral cohomology class in real de Rham cohomology. Equivalently, its periods over integral two-cycles are integers. Some conventions require the class of $\omega /(2\pi )$ to be integral instead; this changes the normalization but not existence.