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In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional plane σ''p'' in the tangent space at ''p''. It is the Gaussian curvature of the surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map at ''p''). The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold. The sectional curvature determines the curvature tensor completely. ==Definition== Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define : Here ''R'' is the Riemann curvature tensor. In particular, if ''u'' and ''v'' are orthonormal, then : The sectional curvature in fact depends only on the 2-plane σ''p'' in the tangent space at ''p'' spanned by ''u'' and ''v''. It is called the sectional curvature of the 2-plane σ''p'', and is denoted ''K''(σ''p''). 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「Sectional curvature」の詳細全文を読む スポンサード リンク
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