Poincaré lemma

In mathematics, Poincaré lemma gives a sufficient condition for a closed differential from to be exact. Precisely, it states that if B is an open ball in Rⁿ, any smooth closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n.

Proof

Translating if necessary, it can be assumed that the ball B has centre 0. Let α_s be the flow on Rⁿ defined by α_s x = e^−s x. For s ≥ 0 it carries B into itself and induces an action on functions and differential forms. The derivative of the flow is the vector field X defined on functions f by Xf = d(α_sf)/ds|_{s = 0}: it is the radial vector field −r ⁠∂/∂r⁠ = −Σ x_i ⁠∂/∂x_i⁠. The derivative of the flow on forms defines the Lie derivative with respect to X given by ${\textstyle L_{X}\omega =\left.{\frac {d(\alpha _{s}\omega )}{ds}}\right|_{s=0}}$. In particular

${\frac {d}{ds}}\alpha _{s}\omega =\alpha _{s}L_{X}\omega ,$

Now define

$h\,\omega =-\int _{0}^{\infty }\alpha _{t}\omega \,dt.$

By the fundamental theorem of calculus we have that

$L_{X}h\,\omega =-\int _{0}^{\infty }\alpha _{t}L_{X}\omega \,dt=-\int _{0}^{\infty }{d \over dt}(\alpha _{t}\omega )\,dt=-[\alpha _{t}\omega ]_{0}^{\infty }=\omega .$

With $\iota _{X}$ being the interior multiplication or contraction by the vector field X, Cartan's formula states that

$L_{X}=d\iota _{X}+\iota _{X}d.$

Using the fact that d commutes with L_X, $\alpha _{s}$ and h, we get:

$\omega =L_{X}h\,\omega =(d\iota _{X}+\iota _{X}d)h\omega =d(\iota _{X}h\omega )+\iota _{X}hd\omega .$

Setting

$g(\omega )=\iota _{X}h(\omega ),$

leads to the identity

$(dg+gd)\,\omega =\omega .$

It now follows that if ω is closed, i. e. dω = 0, then d(g ω) = ω, so that ω is exact and the Poincaré lemma is proved.

(In the language of homological algebra, g is a "contracting homotopy".)

The same method applies to any open set in Rⁿ that is star-shaped about 0, i.e. any open set containing 0 and invariant under α_t for $1<t<\infty$.

Another standard proof of the Poincaré lemma uses the homotopy invariance formula and can be found in Singer & Thorpe (1976, pp. 128–132), Lee (2012), Tu (2011) and Bott & Tu (1982). See Integration along fibers#Example. The local form of the homotopy operator is described in Edelen (2005) and the connection of the lemma with the Maurer-Cartan form is explained in Sharpe (1997).

This formulation can be phrased in terms of homotopies between open domains U in R^m and V in Rⁿ. If F(t,x) is a homotopy from [0,1] × U to V, set F_t(x) = F(t,x). For $\omega$ a p-form on V, define

$g(\omega )=\int _{0}^{1}\iota _{\partial _{t}}(F_{t}^{*}(\omega ))\,dt$

Then

$(dg+gd)\,\omega =\int _{0}^{1}(d\iota _{\partial _{t}}+\iota _{\partial _{t}}d)F_{t}^{*}(\omega )\,dt=\int _{0}^{1}L_{\partial _{t}}F_{t}^{*}(\omega )\,dt=\int _{0}^{1}\partial _{t}F_{t}^{*}(\omega )\,dt=F_{1}^{*}(\omega )-F_{0}^{*}(\omega ).$

Example: In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.

If ω = p dx + q dy is a closed 1-form on (a, b) × (c, d), then p_y = q_x. If ω = df then p = f_x and q = f_y. Set

$g(x,y)=\int _{a}^{x}p(t,y)\,dt,$

so that g_x = p. Then h = f − g must satisfy h_x = 0 and h_y = q − g_y. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

$h(x,y)=\int _{c}^{y}q(a,s)\,ds-g(a,y)=\int _{c}^{y}q(a,s)\,ds,$

and hence

$f(x,y)=\int _{a}^{x}p(t,y)\,dt+\int _{c}^{y}q(a,s)\,ds.$

Similarly, if Ω = r dx ∧ dy then Ω = d(a dx + b dy) with b_x − a_y = r. Thus a solution is given by a = 0 and

$b(x,y)=\int _{a}^{x}r(t,y)\,dt.$

Alternative notions

Related notions to the Poincaré lemma can be proven in other contexts.

On complex manifolds, the use of the Dolbeault operators $\partial$ and ${\bar {\partial }}$ for complex differential forms, which refine the exterior derivative by the formula $d=\partial +{\bar {\partial }}$, lead to the notion of ${\bar {\partial }}$-closed and ${\bar {\partial }}$-exact differential forms. The local exactness result for such closed forms is known as the Dolbeault–Grothendieck lemma (or ${\bar {\partial }}$-Poincaré lemma). Importantly, the geometry of the domain on which a ${\bar {\partial }}$-closed differential form is ${\bar {\partial }}$-exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist counterexamples to the lemma even on contractible domains. The ${\bar {\partial }}$-Poincaré lemma holds in more generality for pseudoconvex domains.

Using both the Poincaré lemma and the ${\bar {\partial }}$-Poincaré lemma, a refined local $\partial {\bar {\partial }}$-Poincaré lemma can be proven, which is valid on domains upon which both the aforementioned lemmas are applicable. This lemma states that $d$-closed complex differential forms are actually locally $\partial {\bar {\partial }}$-exact (rather than just $d$ or ${\bar {\partial }}$-exact, as implied by the above lemmas).

Basic derivation for a 1-form

Let $G\subset \mathbf {R} ^{n}\,(n\geq 2)\,$ be an open star domain and let $\mathbf {a} =(a_{1},...,a_{n})\,$ be a vantage point of $G$, so for all $\mathbf {x} =(x_{1},...,x_{n})\in G\,$ the line segment $[\mathbf {a} ,\mathbf {x} ]$ lies fully in $G$. And let $\,g_{k}:G\to \mathbf {R} ^{m}\,(k=1,...,n)\,$ be continuously differentiable functions. (Possibly some of them are zero.)

Now $\omega =\sum _{j=1}^{n}g_{j}(\mathbf {x} )dx_{j}\,$ is a (differential) 1-form and its exterior derivative is $d\omega =\sum _{j=1}^{n}\sum _{k=1}^{n}{\frac {\partial {g_{j}}}{\partial {dx_{k}}}}dx_{k}\wedge dx_{j}\,$

Since $dx_{j}\wedge dx_{j}=0,\,dx_{j}\wedge dx_{k}=-dx_{k}\wedge dx_{j}$ it follows $d\omega =\sum _{j=1}^{n-1}\sum _{k=j+1}^{n}({\frac {\partial {g_{j}}}{\partial {dx_{k}}}}dx_{k}\wedge dx_{j}+{\frac {\partial {g_{k}}}{\partial {dx_{j}}}}dx_{j}\wedge dx_{k})=\sum _{j=1}^{n-1}\sum _{k=j+1}^{n}({\frac {\partial {g_{k}}}{\partial {dx_{j}}}}-{\frac {\partial {g_{j}}}{\partial {dx_{k}}}})dx_{j}\wedge dx_{k}$

So $d\omega =0\,$ (i.e. $\omega$ is closed) implies:

For all $j,k\in \{1,...,n\}$ it should hold: $\,\,{\frac {\partial g_{k}}{\partial x_{j}}}(\mathbf {x} )={\frac {\partial g_{j}}{\partial x_{k}}}(\mathbf {x} )$

Then $\,f(\mathbf {x} ):=\int _{0}^{1}\sum _{j=1}^{n}\left[g_{j}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot (x_{j}-a_{j})\right]dt\,$ defines a continuously differentiable function $\,G\to \mathbf {R} ^{m}\,$ with $\,{\frac {\partial f}{\partial x_{k}}}(\mathbf {x} )=g_{k}(\mathbf {x} )\,$ for all $k\in \{1,...,n\}$.

The integral is well-defined for all $\mathbf {x} \in G\,$ because $\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\in G\,$ for all $\,t\in [0;1]\,$ and the integrand is a finite sum of continuous functions $t\mapsto g_{j}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot \left(\mathbf {x} -\mathbf {a} \right)$ over the closed real interval $[0;1]$.

Now $\,{\frac {\partial f}{\partial x_{k}}}(\mathbf {x} )=\int _{0}^{1}\sum _{j=1}^{n}{\frac {\partial }{\partial {x_{k}}}}\left[g_{j}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot (x_{j}-a_{j})\right]dt\,=\,$ $\int _{0}^{1}\sum _{j=1}^{n}\left[{\frac {\partial {g_{j}}}{\partial {x_{k}}}}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot t\cdot (x_{j}-a_{j})\,+\{g_{j}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot {\frac {\partial }{\partial {x_{k}}}}\left(x_{j}-a_{j}\right)\}\right]dt\,=\,$ $\int _{0}^{1}\left(g_{k}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)+\sum _{j=1}^{n}\left[{\frac {\partial {g_{k}}}{\partial {x_{j}}}}\left(\mathbf {a} +t\left(\mathbf {x} -\mathbf {a} \right)\right)\cdot t\cdot (x_{j}-a_{j})\right]\right)dt\,=\,$ $\int _{0}^{1}{\frac {d}{dt}}\left[t\cdot g_{k}(\mathbf {a} +t(\mathbf {x} -\mathbf {a} ))\right]dt=g_{k}(\mathbf {x} )$. (Source found at math.stackexchange.)

Relative Poincaré lemma

The relative Poincaré lemma generalizes Poincaré lemma from a point to a submanifold (or some more general closed subset). Let V be a submanifold of a manifold M and U a tubular neighborhood of V. If $\sigma$ is a closed k-form on U that vanishes on V, then there exists a (k-1)-form $\eta$ on U such that $d\eta =\sigma$ and $\eta$ vanishes on V.

References

• Napier, Terrence; Ramachandran, Mohan (2011), An introduction to Riemann surfaces, Birkhäuser, ISBN 978-0-8176-4693-6
• Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer, ISBN 0-387-90894-3