From Poincaré’s Lemma to Volterra’s Theorem

‘A Forgotten Turning‑Point in the History of Differential Forms from this paper of Hans Samelson ‘

 Setting the Scene

Modern paper on differential forms always refer to “early days” usual heroes – Clairaut, Euler, Poincaré, de Rham.
What Samelson found instead was the contribution of the work of the Italian mathematician Vito Volterra (1860 – 1940). In fact, the celebrated Poincaré Lemma appeared a decade earlier, embedded in a trilogy of short notes by Volterra – notes that also anticipate the exterior derivative, the Hodge star, and even the Laplacian on forms.

 Volterra’s Leap Beyond Functions

Volterra entered the story via his study of “funzioni delle linee” – functionals of curves and, later, of higher‑dimensional manifolds in \(\mathbf{R}^n\).
He defined an r‑functional \(\phi\) on closed r‑manifolds, demanded an additivity property: given two manifolds \(S_1,S_2\) we have

where \(S_{12}\) is \(S_1\cup S_2\) with the common submanifold removed. Then he introduced the object we now call the exterior derivative:

where \(\partial S\) is the boundary of an oriented manifold \(S\) of dimension \(r+1\).

From this deceptively simple rule he extracted a full algebra of skew‑symmetric coefficients, observed that \(d^2=0\) (“closedness”), and, crucially, proved both directions of the lemma that a closed form on a star‑shaped domain is exact – decades before the lemma became standard textbook fare.

 Other gems in the Volterra notes

• Hodge star & codifferential. Volterra explicitly defined the \(\ast\) operator

  \[\ast dx^{i_1} \ldots dx^{i_r} = dx^{j_1} \ldots dx^{j_{n-r}}\]

  where the \(i_u\) together with the \(j_v\) form an even permutation of \(1,...,n\). Then \(d^*=\pm \ast d\ast\).
    

• Laplacian identity. He established

  \[d d^*+d^*d=\Delta\]

foreshadowing the modern Hodge theory of harmonic forms.

• Conjugate forms. Seeking an analogue of analytic conjugation, he called two forms \(\omega\) (degree r–1) and \(\kappa\) (degree n–r–1) conjugate when \(\ast d\omega  = d\kappa\), a striking pre‑Yang–Mills viewpoint.

In short, Samelson argues, “Poincaré’s lemma is really Volterra’s theorem.”

(unless some earlier author appears)

 Why Was Volterra Forgotten?

Samelson hints at several reasons: Volterra published in the Rendiconti dei Lincei (less visible to French and German geometers), he phrased everything in the language of functionals rather than forms, and his later political exile (he refused Mussolini’s loyalty oath in 1931 and lost his chair) severed many academic ties.

 Beyond Differential Forms – Volterra’s Wider Mathematical Legacy

For a concise biography see the MacTutor History of Mathematics and the Wikipedia entry on Vito Volterra.

 Take‑aways for Today’s Reader

Re‑reading Volterra shows how ideas sometimes arrive fully formed but linguistically disguised.
Had his “functions of manifolds” been translated into the now‑standard tensor notation a generation earlier, the history of modern geometry might list Volterra next to Cartan and de Rham.

For graduate students the moral is clear: trace ideas to their original papers – and mind the language barrier.

References

1. H. Samelson, Differential Forms, the Early Days; or the Stories of Deahna’s Theorem and of Volterra’s Theorem, Amer. Math. Monthly 108 (2001) 522‑530.
2. V. Volterra, Note sui funzionali delle linee e delle superficie, Rendiconti Accad. Lincei (1889–1890).
3. Vito Volterra – Wikipedia entry (accessed Aug 2025).
4. J.J. O’Connor & E.F. Robertson, Vito Volterra biography, MacTutor History of Mathematics (2009).
5. A. Guerragio & G. Paolini, Vito Volterra (Springer Biographies, 2012).