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Closed differential forms are exact on star-shaped coordinate domains

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open and star-shaped with respect to $a \in U$ , and let $α \in Ω^{k} (U)$ be a smooth closed differential form with $1 \leq k \leq n$ . Define

$(K α)_{x} (v_{1}, \dots, v_{k - 1}) : = \int_{0}^{1} t^{k - 1} α_{a + t (x - a)} (x - a, v_{1}, \dots, v_{k - 1}) d t .$

Then $K α$ is a smooth $(k - 1)$ -form on $U$ and $d (K α) = α$ .

Statement references

Advanced Lectures on General Relativity, §1.3.3

Some Properties of Noether Charge, §4

Facts & Assumptions

Given: The open star-shaped set $U \subseteq R^{n}$ , its centre $a$ , the closed smooth $k$ -form $α$ , and $1 \leq k \leq n$ .

[A1]

$U \subseteq R^{n}$ is open and star-shaped about $a$ , and $α$ is a smooth closed $k$ -form with $1 \leq k \leq n$ .

[F1]

Star-shapedness means $a + t (x - a) \in U$ for every $x \in U$ and $0 \leq t \leq 1$ . Star-shaped open subsets of Euclidean space

[F2]

For a smooth map $F$ , the pullback is $(F^{*} α)_{p} (v_{1}, \dots, v_{k}) = α_{F (p)} (d F_{p} v_{1}, \dots, d F_{p} v_{k})$ . The pullback of a differential form

[F3]

The exterior derivative commutes with pullback: $d (F^{*} α) = F^{*} (d α)$ . The exterior derivative commutes with pullback

[F4]

A continuous parameter derivative on a compact rectangle may be passed through a Riemann integral. Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral

[F5]

For a continuous function on a closed interval with an integrable interior derivative, the integral of that derivative is the endpoint increment. Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative

Proof

Given: As in [A1] and [F1]–[F5].

1.1A1F1F2

Define $F : (0, 1) \times U \to U$ by $F (t, x) = a + t (x - a)$ . By [F1] this map has image in $U$ , and both its domain and target are boundaryless smooth manifolds, so [F3] applies to this restriction. In product coordinates decompose $F^{*} α = γ_{t} + d t \land η_{t}$ , where $γ_{t}$ is a $k$ -form and $η_{t}$ a $(k - 1)$ -form on $U$ . The same affine formula, viewed first as a map $\overset{ˉ}{F} : R \times U \to R^{n}$ , has image in $U$ on a neighborhood of every $(t_{0}, x_{0})$ with $0 \leq t_{0} \leq 1$ , because $\overset{ˉ}{F} (t_{0}, x_{0}) \in U$ and $U$ is open. Thus the coefficient formulas for $γ_{t}$ and $η_{t}$ extend smoothly to $t = 0, 1$ locally in $(t, x)$ .

2.1step 1.1F4

The integrand defining $K α$ is alternating and multilinear in $v_{1}, \dots, v_{k - 1}$ , so its integral is a $(k - 1)$ -covariant alternating tensor at each $x$ . Its coefficient functions are smooth in $(x, t)$ . Restricting $x$ to a compact coordinate rectangle inside $U$ and using [F4] on each coordinate derivative shows that parameter derivatives pass through the $t$ -integral; iteration gives smooth coefficients, hence a smooth $(k - 1)$ -form $K α$ .

2.2A1F2F3step 1.1

Since $d α = 0$ , [F3] gives $d (F^{*} α) = 0$ on $(0, 1) \times U$ . In product coordinates $d = d_{U} + d t \land \partial_{t}$ , so the coefficient of $d t$ is $\partial_{t} γ_{t} - d_{U} η_{t} = 0$ for $0 < t < 1$ . By the local smooth extension in step 1.1, this coefficient identity extends by continuity to $t = 0, 1$ . The pullback formula [F2] gives $(η_{t})_{x} (v_{1}, \dots, v_{k - 1}) = t^{k - 1} α_{a + t (x - a)} (x - a, v_{1}, \dots, v_{k - 1})$ for interior $t$ , and then also at the endpoints by continuity.

3.1step 2.1step 2.2F4F5∎

Thus $K α = \int_{0}^{1} η_{t} d t$ . By step 2.1 and [F4], $d_{U} \int_{0}^{1} η_{t} d t = \int_{0}^{1} d_{U} η_{t} d t$ . Apply [F5] to the coefficient functions of $γ_{t}$ and use step 2.2: $γ_{1} - γ_{0} = \int_{0}^{1} \partial_{t} γ_{t} d t = d_{U} \int_{0}^{1} η_{t} d t$ . Here $γ_{1} = α$ because $F (1, x) = x$ , while $γ_{0} = 0$ because $F (0, x) = a$ is constant and $k \geq 1$ . Hence $d (K α) = α$ .

Remarks

The conclusion is local: each star-shaped coordinate domain is contractible. It supplies local primitives only and gives no global cohomological conclusion.

The cited general-relativity sections use local exactness for their Noether currents; they do not state this general $k$ -form homotopy formula. The displayed argument proves the broader statement here.

Closed differential forms are exact on star-shaped coordinate domains

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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