TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 159, September 1971 INTEGRATION THEORY ON INFINITE-DIMENSIONAL MANIFOLDS BY HUI-HSIUNG KUOC) Abstract. The purpose of this paper is to develop a natural integration theory over a suitable kind of infinite-dimensional manifold. The type of manifold we study is a curved analogue of an abstract Wiener space. Let ii be a real separable Hubert space, B the completion of H with respect to a measurable norm and ;' the inclusion map from H into B. The triple (/, H, B) is an abstract Wiener space. B carries a family of Wiener measures. We will define a Riemann-Wiener manifold to be a triple (#^ t, g) satisfying specific conditions, if" is a C-difTerentiable manifold (y'ä3) modelled on B and, for each x in "W,t(x) is a norm on the tangent space r,(#0 of W at x while g(x) is a densely defined inner product on tJJP). We show that each tangent space is an abstract Wiener space and there exists a spray on ir" associated with g. For each point x in #" the exponential map, defined by this spray, is a C 2-homeomorphism from a r(^)-neighborhood of the origin in Txi'W) onto a neighborhood of x in iK We thereby induce from Wiener measures of TJ¿IP~) a family of Borel measures qt(x, •), f >0, in a neighborhood of x. We prove that qt{x, ■) and qs(y, ■), as measures in their common domain, are equivalent if and only if t = s and d,(x, y) is finite. Otherwise they are mutually singular. Here d, is the almost-metric (in the sense that two points may have infinite distance) on IV deter- mined by g. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals. 0. Introduction. The notion of an abstract Wiener space, first introduced by Gross [8], generalizes that of the classical Wiener space. Wiener [16] put a measure in the Banach space C consisting of the real-valued continuous functions on [0, 1] which vanish at 0 with the sup norm. The subset C consisting of the absolutely continuous functions in C with square integrable derivative forms a Hubert space with the inner product <x, J> = J¿ x'{t)y'{t) dt. Let / be the inclusion map from C into C; then the triple (/, C, C) is known as the classical Wiener space. Received by the editors August 3, 1970. AMS 1969subject classifications. Primary 2846, 6005; Secondary 5750, 5755. Key words and phrases. Integration theory, infinite-dimensional manifold, abstract Wiener space, Wiener measures, Jacobi Theorem, Riemann-Wiener manifold, Riemannian manifold, admissible transformation, spray, exponential map, Radon-Nikodym derivative, equivalence- perpendicularity dichotomy. (') Results are included in the author's doctoral dissertation at Cornell University in 1970. The author wishes to express his deep gratitude to Professor Leonard Gross under whose direction the thesis was written. Copyright © 1971, American Mathematical Society 57 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use