View source for Riemann tensor

''Riemann curvature tensor''

A four-valent tensor that is studied in the theory of curvature of spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820701.png" /> be a space with an affine connection and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820702.png" /> be the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) of the connection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820703.png" />. The components (coordinates) of the Riemann tensor, which is once contravariant and three times covariant, take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820704.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820705.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820706.png" /> is the symbol of differentiation with respect to the space coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820708.png" />. In a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820709.png" /> with a metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207010.png" />, in addition to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207011.png" /> the four times covariant Riemann tensor obtained by lowering the upper index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207012.png" /> using the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207013.png" /> is also studied

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207014.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207015.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207016.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207017.png" /> since the Riemannian connection (without torsion) is considered on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207018.png" />. In an arbitrary space with an affine connection without torsion the coordinates of the Riemann tensor satisfy the first Bianchi identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207019.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207020.png" /></td> </tr></table>

i.e. the cyclic sum with respect to the first three subscripts is zero.

The Riemann tensor possesses the following properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207021.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207022.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207024.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207026.png" />, if both subscripts of one pair are identical, then the corresponding coordinate equals zero: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207027.png" />;

5) the second Bianchi identity is applicable to the absolute derivatives of the Riemann tensor:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207029.png" /> is the symbol for covariant differentiation in the direction of the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207030.png" />. The same identity is applicable to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207031.png" />.

A Riemann tensor has, in all, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207032.png" /> coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207033.png" /> being the dimension of the space, among which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207034.png" /> are essential. Between the latter no additional dependencies result from the properties listed above.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207035.png" /> the Riemann tensor has one essential coordinate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207036.png" />; it forms part of the definition of the intrinsic, or Riemannian, curvature of the surface: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207037.png" /> (see [[Gaussian curvature|Gaussian curvature]]).

The Riemann tensor was defined by B. Riemann in 1861 (published in 1876).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Eisenhart,   "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Gromoll,   W. Klingenberg,   W. Meyer,   "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1–2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,   "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Schouten,   D.J. Struik,   "Einführung in die neueren Methoden der Differentialgeometrie" , '''2''' , Noordhoff  (1924)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.P. Eisenhart,   "An introduction to differential geometry with the use of the tensor calculus" , Princeton Univ. Press  (1947)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>