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 H H be a real separable Hilbert space, B B the completion of H H with respect to a measurable norm and i i the inclusion map from H H into B B . The triple ( i , H , B ) (i,H,B) is an abstract Wiener space. B B carries a family of Wiener measures. We will define a Riemann-Wiener manifold to be a triple ( W , τ , g ) (\mathcal {W},\tau ,g) satisfying specific conditions, W \mathcal {W} is a C j {C^j} -differentiable manifold ( j ≧ 3 ) (j \geqq 3) modelled on B B and, for each x x in W , τ ( x ) \mathcal {W},\tau (x) is a norm on the tangent space T x ( W ) {T_x}(\mathcal {W}) of W \mathcal {W} at x x while g ( x ) g(x) is a densely defined inner product on T x ( W ) {T_x}(\mathcal {W}) . We show that each tangent space is an abstract Wiener space and there exists a spray on W \mathcal {W} associated with g g . For each point x x in W \mathcal {W} the exponential map, defined by this spray, is a C j − 2 {C^{j - 2}} -homeomorphism from a τ ( x ) \tau (x) -neighborhood of the origin in T x ( W ) {T_x}(\mathcal {W}) onto a neighborhood of x x in W \mathcal {W} . We thereby induce from Wiener measures of T x ( W ) {T_x}(\mathcal {W}) a family of Borel measures q t ( x , ⋅ ) , t > 0 {q_t}(x, \cdot ),t > 0 , in a neighborhood of x x . We prove that q t ( x , ⋅ ) {q_t}(x, \cdot ) and q s ( y , ⋅ ) {q_s}(y, \cdot ) , as measures in their common domain, are equivalent if and only if t = s t = s and d g ( x , y ) {d_g}(x,y) is finite. Otherwise they are mutually singular. Here d g {d_g} is the almost-metric (in the sense that two points may have infinite distance) on W \mathcal {W} determined by g g . In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals.