Introduction
In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds to develop his General Theory of Relativity. In particular, his equations for gravitation are restrictions on the curvature of space.
Read more about this topic: Riemannian Manifold
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