ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR QUARTERLY REPORT NO. 3 AFFINE STRUCTURES IN EUCLIDEAN.MANIFOLDS December 2, 1952, to March 1, 1953 By EDWIN E. MOISE Assistant Professor of Mathematics Project 2051 ORDNANCE CORPS, DEPARTMENT OF THE ARMY CONTRACT DA-20-018-ORD-12277 March, 1953

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN AFFINE STRUCTURES IN EUCLIDEAN MANIFOLDS Substantial simplifications have been made in the proof of Theorem 3 of Affine structures in 3-manifolds, I. Polyhedral approximations of solids, Annals of Mathematics, vol. 54 (1941), pp. 506-533. This paper consists, essentially, of one very long and intricate proof; a simplified version of such a proof appears to be worthwhile. The revised and simplified version is now about half completed. Probably it will be published, although it contains no new results. Most of the chief investigator's time has been spent in efforts to solve certain of the problems described in the original proposal. But these efforts have not so far led to any results worth reporting.