The theorema egregium or, in essence, the fundamental theorem of riemannian geometry asserts that curvature is an invariant of the metric. We ask the converse: how far does curvature determine the metric? Important theorems in this direction are the classical theorems for (embedded) surfaces. More recently there is a local theorem of E. Cartan and its global formulation due to W. Ambrose (cf. [1]). For a different approach see Nomizu and Yano [8]. In these theorems there are non-trivial hypotheses about the curvature tensor. We ask a more naive, but geometrically fascinating question: let (M, g), (M, U) be two Riemann manifolds. Denote the corresponding sectional curvatures by K respectively K. We say, M, M are isocurved if there exists a "sectional-curvature-preserving" diffeomorphism f: M O M, i.e., for every p e M and for every a, a 2-plane section of the tangent space Tp(M), we have K(a) = K(f* a) .