Writting a report on the Ph.D. thesis of Paolo Valentini I had to learn what are Lipschitz-Killing forms. In particular, the author uses Lemma 3.1 from the paper Some remarks on Legendrian rectifiable currents by Joseph Fu. Later, I learned there was an erratum to that paper but Lemma 3.1 was not affected.

The said lemma provides a formula for the exterior derivative of \(j^{\mathrm{th}}\) Lipschitz-Killing form in terms of lower \((j-1)^{\mathrm{th}}\) Lipschitz-Killing form and the contact form. I could not understand the proof since it uses Maurer-Cartan equations with which I am not acquainted. In result, I wrote my own proof which works only in the Euclidean setting but is done with elementary tools.