Let $\alpha$ be an exterior product of a harmonic and a parallel form on a Riemannian manifold. Then $\alpha$ is known to be harmonic. I have heard that this is an old result due to R. Bott, but I could never find a reference. I would be very grateful for any pointers to the early literature.



reference to a theorem about a product of harmonic and parallel forms
One place where (a generalization of) the desired result is stated explicitly is in a 1973 paper by J. H. Sampson, [On a theorem of Chern](https://www.jstor.org/stable/1996588). Sampson gives a simplified proof of Chern's main result in his 1957 paper *On a generalization of Kähler geometry* (Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 103–121. Princeton University Press, Princeton, N. J., 1957).


There is also a very nice 1962 exposition (in French) of Chern's theorem by André Weil, [Un théorème fondamental de Chern en géométrie riemannienne](http://www.numdam.org/item/SB_1961-1962__7__273_0.pdf), in *Séminaire N. Bourbaki, 1962, exp. no 239, p. 273-284*.


The form in which Sampson and Weil state Chern's result is as follows:


*Theorem:* If a Riemannian manifold $(M^n,g)$ has holonomy $H\subseteq\mathrm{O}(n)$, then the $g$-Laplacian commutes with all of the linear operators on $\Omega^\*(M)$ constructed from the ring of $H$-equivariant linear maps $L:\Lambda^\*(\mathbb{R}^n)\to\Lambda^\*(\mathbb{R}^n)$.


The desired result is a special case of this theorem, since, if $\pi$ is a $g$-parallel $p$-form, then the holonomy $H$ of $g$ is contained in the stabilizer of $\pi$, and hence, by the above result, the operator $L(\alpha) = \alpha\wedge\pi$ commutes with the Laplacian of $g$. Of course, this implies the desired result since $\Delta(\alpha\wedge\pi) = \Delta\bigl(L(\alpha)\bigr) = L\bigl(\Delta(\alpha)\bigr) = 0$ when $\Delta(\alpha) = 0$, but is stronger.


*Comments:* Chern proved the above theorem in his 1957 paper for the ring of operators $L$ as above that are degree-preserving, but, in fact, this implies the more general result.


Both Chern and Sampson phrase the theorem in terms of $H$-structures without torsion, but, of course, this is the same as Riemannian manifolds with holonomy contained in $H$, a point explicitly made by Weil.


Sampson does not refer to Weil's article; perhaps he was unaware of it. Meanwhile, Sampson goes on to apply Chern's idea to other Laplacians and proves new results.


There are earlier results in special cases by André Lichnerowicz, e.g. *Généralisations de la géométrie kählérienne globale.* (French) Colloque de Géométrie Différentielle, Louvain, 1951, pp. 99–122. Georges Thone, Liège; Masson & Cie., Paris, 1951, but neither his articles nor those of Chern, Sampson, or Weil mention Bott.


