The Einstein field equation in a multidimensional universe (Q1101717)

String theory [\textit{M. B. Green}, \textit{J. H. Schwarz} and \textit{E. Witten}, Superstring theory: 1 and 2 (1987; Zbl 0619.53002)] predicts that the universe has 10 or 26 dimensions. A salient problem is how the Einstein field equation should be written in terms of these revivified Kaluza-Klein cosmologies. The answer is by now well-known, yet nobody seems to have rewritten the seminal computation in [\textit{D. Lovelock}, J. Math. Phys. 12, 498-501 (1971; Zbl 0213.488)] where an unnecessarily involved Euler-Lagrange variational method is employed and, curiously enough, no allusion to the Gauss-Bonnet-Chern theorem is made. We provide a more straightforward argument, which has been inspired by Hilbert's original derivation of the Einstein field equation.