Space Form
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High Quality Content by WIKIPEDIA articles! In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic. The possible extensions are limited. One might wish to conjecture that the manifolds are isometric, but rescaling the Riemannian metric on a compact…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic. The possible extensions are limited. One might wish to conjecture that the manifolds are isometric, but rescaling the Riemannian metric on a compact aspherical Riemannian manifold preserves the fundamental group and shows this to be false. One might also wish to conjecture that the manifolds are diffeomorphic, but John Milnor's exotic spheres are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.