Riemann curvature tensor

Let 𝒳 denote the vector spaceMathworldPlanetmath of smooth vector fields on a smooth Riemannian manifold (M,g). Note that 𝒳 is actually a 𝒞(M) module because we can multiply a vector field by a function to obtain another vector field. The Riemann curvature tensorMathworldPlanetmath is the tri-linear 𝒞 mapping


which is defined by


where X,Y,Z𝒳 are vector fields, where is the Levi-Civita connectionMathworldPlanetmath attached to the metric tensor g, and where the square brackets denote the Lie bracket of two vector fields. The tri-linearity means that for every smooth f:M we have


In componentsPlanetmathPlanetmathPlanetmath this tensor is classically denoted by a set of four-indexed components Rijkl. This means that given a basis of linearly independentMathworldPlanetmath vector fields Xi we have


In a two dimensional manifold it is known that the Gaussian curvatureMathworldPlanetmath it is given by

Title Riemann curvature tensor
Canonical name RiemannCurvatureTensor
Date of creation 2013-03-22 16:26:17
Last modified on 2013-03-22 16:26:17
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 10
Author juanman (12619)
Entry type Definition
Classification msc 53B20
Classification msc 53A55
Related topic CurvaturePlanetmathPlanetmath
Related topic Connection
Related topic FormalLogicsAndMetaMathematics