50 R. S. PALAIS [February manifold. Thus a cartesian factor of a homotopy manifold is not necessarily a homotopy manifold. References 1. R. Bing, A decomposition of E3 into points and tame arcs such that the decomposi- tion space is topologically different from E3, Ann. of Math. vol. 65 (1957). 2. M. Curtis and R. Wilder, The existence of certain types of manifolds, Trans. Amer. Math. Soc. vol. 91 (1959) pp. 152-160. 3. H. Griffiths, A contribution to the theory of manifolds, Michigan Math. J. vol. 2 (1953) pp. 61-89. 4. R. Roxen, E* is the Cartesian product of a totally non-euclidean space and E1, Notices Amer. Math. Soc, Abstr. 563-1, vol. 6 (1959) p. 641. 5. S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. vol. 8 (1957) pp. 604-610. 6. R. Wilder, Monotonemappings of manifolds, Pacific J. Math. vol. 3 (1957) pp. 1519-1528. 7. -, Monotone mappings of manifolds II, Michigan Math. J. vol. 5 (1958) pp. 19-25. The Seoul National University, Seoul, Korea LOGARITHMICALLY EXACT DIFFERENTIAL FORMS RICHARD S. PALAIS1 Let M be a connected, differentiable ( = Cx) manifold. Let Cl(M, C) denote the complex vector space of complex valued one-forms on M: an element w of Cl(M, C) is a function which assigns to each x£M a linear map wx of Mx (the tangent space to M at x, a real vector space) into the complex numbers C, such that if X is a differ- entiable vector field on M then x—xax(Xx) is a differentiable complex valued function on M. Each element u of CX(M, C) can be written uniquely in the form ß+iv where ß and v are real valued one-forms on M, and we put ¿i=Reco and w< = Imco. We write Zl(M, C) for the subspace of C1(M, C) consisting of closed forms and Bl{M, C) for the subspace of Zl(M, C) consisting of exact forms. An element of OiM, C) will be called logarithmically exact if it is of the form df/f for some nowhere vanishing, differentiable, complex valued function / on M. Since d(df/f) = (fd2f-dfAdf)/f2 = 0 and df/f~dg/g = d(f/g)/(f/g) it is clear that the set L^M, C) of logarith- mically exact one-forms is a subgroup (but not in general a subspace) Received by the editors February 24, 1960. 1 The author is a National Science Foundation postdoctoral fellow.