Tensor of Riemann

In differential Geometry, the tensor of curve of Riemann is the most current way to express the Courbure varieties riemanniennes, or more general of a variety having a Connection refines, with or without Torsion.

That is to say two Geodetic S of a space curve, parallel S in the vicinity of a point P . In a curved space, geodetic parallels in a point inevitably will not remain it in other points of space. The tensor of curve of Riemann expresses the evolution of these geodetic one compared to the other. The curved space is, the more the geodetic ones will approach or move away quickly .

Definition

The tensor of curve is formulated using a Connection of Levi-Civita (or more generally of a connection refines) $\backslash \; nabla$ (or Dérivée covariante) by the following formula:

For any vector U , v and W of the variety:

where $,\; \backslash$ is the Crochet of Dregs.

Here $R\; (U,\; v)$ is a linear transformation according to each one of its arguments on the tangent space of the variety.

NB : certain authors define the tensor of curve like opposite sign.

If $u=\; \{\backslash \; partial\; \backslash \; over\; \backslash \; partial\; x\_i\}$ and $v=\; \{\backslash \; partial\; \backslash \; over\; \backslash \; partial\; x\_j\}$ is fields of vector of coordinates, then $=0$ and one can rewrite the formula:

$R\; (U,\; v)\; w=\; \backslash \; nabla\_u\; \backslash \; nabla\_v\; W\; -\; \backslash \; nabla\_v\; \backslash \; nabla\_u\; W$

The tensor of curve then measures the not-commutation of derived the covariante .

The linear transformation $w\; \backslash \; mapsto\; R\; (U,\; v)\; w$ is also called the tranformation of curve or endomorphism .

In term of coordinates, this equation can be rewritten by using the Symboles of Christoffel:

Symmetries and identities

The tensor of curve of Riemann has following symmetries:

$R\; (U,\; v)\; =-R\; (v,\; U)\; \_\; \{\}\; ^\; \{\}$

$\backslash \; langle\; R\; (U,\; v)\; W,\; Z\; \backslash \; rangle=-\; \backslash \; langle\; R\; (U,\; v)\; Z,\; W\; \backslash \; rangle^\; \{\}\; \_\; \{\}$

$R\; (U,\; v)\; w+R\; (v,\; W)\; u+R\; (W,\; U)\; v=0\; ^\; \{\}\; \_\; \{\}$

The last identity was discovered by Ricci, but is often named first identity of Bianchi or algebraic identity of Bianchi .

These three identities form a complete listing of symmetries of tensor of curve, i.e. being given a tensor respecting the identities above, one can find a variety of Riemann having such a tensor of curve in a point. Simple mathematical calculations show that such a tensor has $n^2\; (n^2-1)\; /12$ component independent.

It is possible to deduce another useful identity starting from these three equations:

$\backslash \; langle\; R\; (U,\; v)\; W,\; Z\; \backslash \; rangle=\; \backslash \; langle\; R\; (W,\; Z)\; U,\; v\; \backslash \; rangle^\; \{\}\; \_\; \{\}$

The identity of Bianchi (often called second identity of Bianchi or differential identity of Bianchi ) implies the derivative covariantes:

$\backslash \; nabla\_uR\; (v,\; W)\; +\; \backslash \; nabla\_vR\; (W,\; U)\; +\; \backslash \; nabla\_w\; R\; (U,\; v)\; =\; 0$

Being given a reference frame given in a point of a variety, the preceding identities can be written in term of the components of the tensor of Riemann like:

$R\_\{abcd\}^\{\}=-R\_\{bacd\}=-R\_\{abdc\}$

$R\_\{abcd\}^\{\}=R\_\{cdab\}$

$6R\; \_\; \{has\}\; ^\; \{\}\; =0$ (first identity of Bianchi)

$6R\; \_\; \{ab\}\; ^\; \{\}\; =0$ (second identity of Bianchi)

where the hooks correspond to the Crochet of Dregs, qualifying symmetrizations according to the indices, and the semicolon represents the Dérivée covariante.

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