TheoremStatement: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)

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Local two-form potentials and global obstruction

Physical scope

Pure mathematics; no physical premises or physical claim.

Statement

Let $U \subseteq R^{4}$ be open and star-shaped about $a \in U$ , meaning $a + t (x - a) \in U$ for every $x \in U$ and $t \in [0, 1]$ . For a smooth closed two-form $F = \frac{1}{2} F_{μ ν} d x^{μ} \land d x^{ν}$ on $U$ , define the smooth one-form $A = A_{μ} d x^{μ}$ by

$A_{μ} (x) = \int_{0}^{1} t (x - a)^{ν} F_{ν μ} (a + t (x - a)) d t .$

Then $d A = F$ . On any smooth manifold $M$ , possibly with boundary, a smooth closed two-form $F$ has a globally defined smooth one-form potential $A$ with $d A = F$ if and only if its de Rham class $[F] \in H_{d R}^{2} (M)$ is zero. If a finite smooth singular two-cycle $z$ has nonzero period $\int_{z} F$ , then $F$ is not globally exact.

Statement references

Lorentzian Hodge star and periods

Closed and exact differential forms

De rham cohomology

The local coordinate formula for the exterior derivative

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

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$

Stokes theorem for smooth singular chains

Facts & Assumptions

Given: The open star-shaped set $U$ , its centre $a$ , a smooth closed two-form $F$ on $U$ , and, for the global clauses, a smooth manifold $M$ , a smooth closed two-form on $M$ , and a finite smooth singular two-cycle $z$ when a period is considered.

[A1]

Star-shapedness is the explicit segment condition in the Statement; closedness means $d F = 0$ , with $F \in Ω^{2}$ smooth.

[F1]

Lorentzian Hodge star and periods fixes $F = \frac{1}{2} F_{μ ν} d x^{μ} \land d x^{ν}$ , the exactness convention, and the period $\int_{z} F$ for finite smooth singular cycles.

[F2]

Closed and exact differential forms defines $Z^{2} = \ker d$ and $B^{2} = im d$ ; in particular $F$ is closed exactly when $d F = 0$ , and exact exactly when $F = d A$ .

[F3]

The local coordinate formula for the exterior derivative gives $(d F)_{λ μ ν} = \partial_{λ} F_{μ ν} + \partial_{μ} F_{ν λ} + \partial_{ν} F_{λ μ}$ .

[F4]

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral permits differentiating a Riemann integral in one parameter when the integrand and its parameter derivative extend continuously to a compact rectangle.

[F5]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative gives $\int_{0}^{1} G^{'} (t) d t = G (1) - G (0)$ for continuous $G$ with a Riemann-integrable interior derivative.

[F6]

De rham cohomology defines $H_{d R}^{2} (M) = Z^{2} (M) / B^{2} (M)$ , so for closed $F$ , $[F] = 0$ exactly when $F$ is globally exact.

[F7]

Stokes theorem for smooth singular chains gives $\int_{z} d A = \int_{\partial z} A$ for smooth $A$ on a possibly boundary manifold and finite smooth singular chains $z$ of positive degree.

[F8]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ gives the derivative of a smooth composition along the segment $x_{t} = a + t (x - a)$ .

[F9]

Proof

1.1A1F1F4given

For each $x \in U$ , the segment $x_{t} = a + t (x - a)$ lies in $U$ by [A1], so the displayed integrand is smooth on $0 \leq t \leq 1$ . The segment is compact; since $U$ is open, it has a neighborhood still contained in $U$ . Varying one coordinate of $x$ over a sufficiently small closed interval gives a compact rectangle on which the integrand and each coordinate derivative are continuous. Applying [F4] successively to these derivatives proves that every $A_{μ}$ is smooth; the formula has no singularity at $t = 0$ .

1.2A1F3F4algebra

Put $y = x - a$ and $x_{t} = a + t y$ . Differentiating under the integral as in [F4] gives $\partial_{μ} A_{ν} = \int_{0}^{1} (t F_{μ ν} (x_{t}) + t^{2} y^{ρ} \partial_{μ} F_{ρ ν} (x_{t})) d t$ . Subtracting the expression with $μ, ν$ exchanged, the coordinate formula [F3] gives $\partial_{μ} F_{ρ ν} + \partial_{ρ} F_{ν μ} + \partial_{ν} F_{μ ρ} = 0$ . Since $F_{ν μ} = - F_{μ ν}$ and $F_{μ ρ} = - F_{ρ μ}$ , this yields $\partial_{μ} F_{ρ ν} - \partial_{ν} F_{ρ μ} = \partial_{ρ} F_{μ ν}$ ; hence $\partial_{μ} A_{ν} - \partial_{ν} A_{μ} = \int_{0}^{1} (2 t F_{μ ν} (x_{t}) + t^{2} y^{ρ} \partial_{ρ} F_{μ ν} (x_{t})) d t$ .

1.3F2F6given

For any closed $F$ , [F6] identifies $[F] = 0$ with $F \in B^{2} (M)$ , which by [F2] means exactly that some globally defined smooth one-form $A$ satisfies $d A = F$ .

2.1F1F5F8F9step 1.2algebra

By [F8] and [F9], the integrand in step 1.2 is $\frac{d}{d t} (t^{2} F_{μ ν} (x_{t}))$ . The function $G (t) = t^{2} F_{μ ν} (x_{t})$ is continuous on $[0, 1]$ , differentiable on $(0, 1)$ , and has continuous derivative there extending to the endpoints, so [F5] gives $\partial_{μ} A_{ν} - \partial_{ν} A_{μ} = F_{μ ν} (x)$ . This is precisely $d A = F$ under the component convention in [F1].

3.1F1F7given∎

If $F = d A$ and $z$ is a finite smooth singular cycle, then [F7] gives $\int_{z} F = \int_{z} d A = \int_{\partial z} A = 0$ , since $\partial z = 0$ . Therefore a nonzero period precludes exactness.

Local two-form potentials and global obstruction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources