ISRAEL JOURNAL OF MATHEMATICS 93 (1996), 157-170 THE EXTERIOR DERIVATIVE AS A KILLING VECTOR FIELD* BY J. MONTERDE Dept. de Geometric i Topologia, Facultat de Matemdtiques, Universitat de Valencia C/Dr. Moliner 50, ~{6100-Burjassot (Valencia), Spain e-mail: monterde@iluso.ci.uv.es AND O. A. S~,NCHEZ-VALENZUELA Centro de Investigacign en Matemdticas Apdo. Postal, 402; C.P. 36000, Guanajuato, Gto., Mdxico e-mail: saval@servidor.dgsca.unam.mx ABSTRACT Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms. 1. Introduction Graded manifold theory, as developed for example in [4], provides a natural framework to address some geometrical questions that arose from the study of the de Rham complex of differential forms on a smooth manifold M. If M is an n-dimensional smooth manifold, and ~t(M) is its corresponding Z2-graded- commutative ]~-algebra of differential forms, the pair (M, ~(M)) is an (n, n)- dimensional Z2-graded manifold (graded manifold for short). Abstractly, graded * Partially supported by DGICYT grants #PB91-0324, and SAB94-0311; CONACyT grant #3189-E9307. Received October 20, 1994 157