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In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (''M'',''g'') is a real smooth manifold ''M'' equipped with an inner product on the tangent space at each point that varies smoothly from point to point in the sense that if ''X'' and ''Y'' are vector fields on ''M'', then is a smooth function. The family of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry. A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. ==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 constraints on the curvature of space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemannian manifold」の詳細全文を読む スポンサード リンク
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