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