PlanetMath: Riemann curvature tensor

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Riemann curvature tensor

(Definition)

Let $\mathcal{X}$ denote the vector space of smooth vector fields on a smooth Riemannian manifold . Note that $\mathcal{X}$ is actually a $\mathcal{C}^\infty(M)$ module because we can multiply a vector field by a function to obtain another vector field. The Riemann curvature tensor is the tri-linear $\mathcal{C}^\infty$ mapping

$\displaystyle R:{\mathcal{X}}\times{\mathcal{X}}\times{\mathcal{X}}\to{\mathcal{X}},$

which is defined by

$\displaystyle R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$

where $X,Y,Z\in\mathcal{X}$ are vector fields, where $\nabla$ is the Levi-Civita connection attached to the metric tensor

, and where the square brackets denote the Lie bracket of two vector fields. The tri-linearity means that for every smooth $f\colon M\to\mathbb{R}$ we have

$\displaystyle fR(X,Y)Z=R(fX,Y)Z=R(X,fY)Z=R(X,Y)fZ.$

In components this tensor is classically denoted by a set of four-indexed components ${R^i}_{jkl}$ . This means that given a basis of linearly independent vector fields we have

$\displaystyle R(X_j,X_k)X_l=\sum_s {R^s}_{jkl}X_s.$

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

$\displaystyle K_g=\frac{R_{1212}}{g_{11}g_{22}-{g_{12}}^2}$

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"Riemann curvature tensor" is owned by juanman. [ full author list (3) ]

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See Also: curvature, connection

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Cross-references: Gaussian curvature, manifold, linearly independent, basis, tensor, components, Lie bracket, square, metric tensor, Levi-Civita connection, mapping, function, module, Riemannian manifold, vector fields, smooth, vector space
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This is version 7 of Riemann curvature tensor, born on 2006-11-28, modified 2007-01-29.
Object id is 8592, canonical name is RiemannCurvatureTensor.
Accessed 1781 times total.

Classification:

AMS MSC:	53A55 (Differential geometry :: Classical differential geometry :: Differential invariants , geometric objects)
	53B20 (Differential geometry :: Local differential geometry :: Local Riemannian geometry)

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