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In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity. In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action. The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics. The scalar curvature is defined as the trace of the Ricci tensor, and it can be characterized as a multiple of the average of the sectional curvatures at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive mass theorem of Richard Schoen, Shing-Tung Yau and Edward Witten. Another is the Yamabe problem, which seeks extremal metrics in a given conformal class for which the scalar curvature is constant. ==Definition== The scalar curvature is usually denoted by ''S'' (other notations are ''Sc'', ''R''). It is defined as the trace of the Ricci curvature tensor with respect to the metric: : The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first raise an index to obtain a (1,1)-valent tensor in order to take the trace. In terms of local coordinates one can write : where ''R''''ij'' are the components of the Ricci tensor in the coordinate basis: : Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows : where are the Christoffel symbols of the metric, and is the partial derivative of in the -th coordinate direction. Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally defined for any affine connection, the scalar curvature requires a metric of some kind. The metric can be pseudo-Riemannian instead of Riemannian. Indeed, such a generalization is vital to relativity theory. More generally, the Ricci tensor can be defined in broader class of metric geometries (by means of the direct geometric interpretation, below) that includes Finsler geometry. 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「Scalar curvature」の詳細全文を読む スポンサード リンク
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