Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described It is one of many things named after Bernhard Riemann and Elwin Bruno Christoffel. Elwin Bruno Christoffel ( November 10, 1829 in Montjoie now called Monschau – March 15, 1900 in Strasbourg) was a The curvature tensor is given in terms of a Levi-Civita connection by the following formula:

$R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]} w .$

Here $R (u, v)$ is a linear transformation of the tangent space of the manifold; it is linear in each argument. In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i

NB. Some authors define the curvature tensor with the opposite sign.

If $u=\partial/\partial x^i$ and $v=\partial/\partial x^j$ are coordinate vector fields then $[u, v] = 0$ and therefore the formula simplifies to

$R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w$

i. e. the curvature tensor measures noncommutativity of the covariant derivative.

The linear transformation $w\mapsto R(u,v)w$ is also called the curvature transformation or endomorphism.

The Riemann curvature tensor, especially in its coordinate expression (see below), is a central mathematical tool of general relativity, the modern theory of gravity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another

1 Coordinate expression
2 Symmetries and identities
3 For surfaces
4 See also

Coordinate expression

In local coordinates $x μ$ the Riemann curvature tensor is given by

${R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})$

where $\partial_{\mu} = \partial/\partial x^{\mu}$ are the coordinate vector fields. The above expression can be written using Christoffel symbols:

${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$

(see also the list of formulas in Riemannian geometry). In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita This is a list of Formulas encountered in Riemannian geometry.

The transformation of a vector $V μ$ after circling an infinitesemal rectangle $d x ν d x σ$ is: $\delta V^\mu = R^\mu_{\nu\sigma\tau} dx^\nu dx^\sigma V^\tau$ .

Symmetries and identities

The Riemann curvature tensor has the following symmetries:

$R(u,v)=-R(v,u)^{}_{}$

$\langle R(u,v)w,z \rangle=-\langle R(u,v)z,w \rangle^{}_{}$

$R(u,v)w+R(v,w)u+R(w,u)v=0 ^{}_{}.$

The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. Gregorio Ricci-Curbastro ( January 12, 1853 - August 6, 1925) was an Italian Mathematician. These three identities form a complete list of symmetries of the curvature tensor, i. e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has $n 2 (n 2 - 1) / 12$ independent components.

Yet another useful identity follows from these three:

$\langle R(u,v)w,z \rangle=\langle R(w,z)u,v \rangle^{}_{}.$

The Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) involves the covariant derivative:

$\nabla_uR(v,w)+\nabla_vR(w,u)+\nabla_w R(u,v) = 0.$

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

$R_{abcd}^{}=-R_{bacd}=-R_{abdc}$

$R_{abcd}^{}=R_{cdab}$

$R_{a[bcd]}^{}=0$ (first Bianchi identity)

$R_{ab[cd;e]}^{}=0$ (second Bianchi identity)

where the square brackets denote cyclic symmetrisation over the indices and the semi-colon is a covariant derivative. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how

For surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

$R_{abcd}^{}=K(g_{ac}g_{db}- g_{ad}g_{cb} )$

where $g a b$ is the metric tensor and $K$ is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures As expected we see that the Riemann curvature tensor only has one independent component.

The Gaussian curvature coincides with the sectional curvature of the surface. In Riemannian geometry, the sectional curvature is one of the ways to describe the Curvature of Riemannian manifolds. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

$\operatorname{Ric}_{ab} = Kg_{ab}.$

Contents

Coordinate expression

Symmetries and identities

For surfaces

See also