Editing Riemann curvature tensor (section)

== Definition ==
Let <math>(M, g)</math> be a [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]], and <math>\mathfrak{X}(M)</math> be the space of all [[Vector field|vector fields]] on <math>M</math>. We define the '''Riemann curvature tensor''' as a map <math>\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)</math> by the following formula{{sfn|Lee|2018|p=196}} where <math>\nabla</math> is the [[Levi-Civita connection]]:
: <math>R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z</math>
or equivalently
: <math>R(X, Y) = [\nabla_X,\nabla_Y] - \nabla_{[X, Y]} </math>
where <math>[X,Y]</math> is the [[Lie bracket of vector fields]] and <math>[\nabla_X,\nabla_Y] </math> is a commutator of differential operators.  It turns out that the right-hand side actually only depends on the value of the vector fields <math>X, Y, Z</math> at a given point, which is notable since the covariant derivative of a vector field also depends on the field values in a neighborhood of the point. Hence, <math>R</math> is a <math>(1,3)</math>-tensor field. For fixed <math>X,Y</math>, the linear transformation <math>Z \mapsto R(X, Y)Z</math> is also called the ''curvature transformation'' or ''endomorphism''. Occasionally, the curvature tensor is defined with the opposite sign.

The curvature tensor measures ''noncommutativity of the covariant derivative'', and as such is the [[integrability condition|integrability obstruction]] for the existence of an isometry with Euclidean space (called, in this context, ''flat'' space).  

Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the [[second covariant derivative]]<ref name="lawson">{{cite book |last1=Lawson |first1=H. Blaine Jr. |last2=Michelsohn |first2=Marie-Louise |author2-link=Marie-Louise Michelsohn |title=Spin Geometry |url=https://archive.org/details/spingeometry00laws |url-access=limited |year=1989 |page=[https://archive.org/details/spingeometry00laws/page/n166 154] |publisher=Princeton U Press |isbn=978-0-691-08542-5}}</ref>
: <math display="inline">\nabla^2_{X,Y} Z = \nabla_X\nabla_Y Z - \nabla_{\nabla_X Y}Z </math>
which depends only on the values of <math>X, Y</math> at a point.
The curvature can then be written as
: <math>R(X, Y) = \nabla^2_{X,Y} - \nabla^2_{Y,X} </math>
Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. In [[abstract index notation]], <math display="block">R^d{}_{cab} Z^c = \nabla_a \nabla_b Z^d - \nabla_b \nabla_a Z^d . </math>The Riemann curvature tensor is also the [[commutator]] of the covariant derivative of an arbitrary covector <math>A_{\nu}</math> with itself:<ref>{{cite book |author=Synge J.L., Schild A. |url=https://archive.org/details/tensorcalculus00syng/page/83 |title=Tensor Calculus |publisher=first Dover Publications 1978 edition |year=1949 |isbn=978-0-486-63612-2 |pages=[https://archive.org/details/tensorcalculus00syng/page/83 83; 107]}}</ref><ref>{{cite book |author=P. A. M. Dirac |title=General Theory of Relativity |publisher=[[Princeton University Press]] |year=1996 |isbn=978-0-691-01146-2}}</ref>
: <math>A_{\nu;\rho\sigma} - A_{\nu;\sigma\rho} = A_{\beta} R^{\beta}{}_{\nu\rho\sigma}.</math>

This formula is often called the ''Ricci identity''.<ref name="lovelockrund">{{cite book |last1=Lovelock |first1=David |title=Tensors, Differential Forms, and Variational Principles |last2=Rund |first2=Hanno |publisher=Dover |year=1989 |isbn=978-0-486-65840-7 |page=84,109 |author-link2=Hanno Rund |orig-year=1975}}</ref> This is the classical method used by [[Gregorio Ricci-Curbastro|Ricci]] and [[Tullio Levi-Civita|Levi-Civita]] to obtain an expression for the Riemann curvature tensor.<ref>{{citation |last1=Ricci |first1=Gregorio |title=Méthodes de calcul différentiel absolu et leurs applications |date=March 1900 |url=https://zenodo.org/record/1428270 |journal=Mathematische Annalen |volume=54 |issue=1–2 |pages=125–201 |doi=10.1007/BF01454201 |s2cid=120009332 |last2=Levi-Civita |first2=Tullio |author-link=Gregorio Ricci-Curbastro}}</ref> This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows <ref>{{cite journal |vauthors=Sandberg, Vernon D |date=1978 |title=Tensor spherical harmonics on S 2 and S 3 as eigenvalue problems |url=https://authors.library.caltech.edu/32877/1/SANjmp78.pdf |journal=[[Journal of Mathematical Physics]] |volume=19 |issue=12 |pages=2441–2446 |bibcode=1978JMP....19.2441S |doi=10.1063/1.523649}}</ref>
: <math>\begin{align}
      &\nabla_\delta \nabla_\gamma T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \nabla_\gamma \nabla_\delta T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} \\[3pt]
  ={} &R^{\alpha_1}{}_{\rho\delta\gamma} T^{\rho\alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} + \ldots +
       R^{\alpha_r}{}_{\rho\delta\gamma} T^{\alpha_1 \cdots \alpha_{r-1}\rho}{}_{\beta_1 \cdots \beta_s} -
       R^{\sigma}{}_{\beta_1\delta\gamma} T^{\alpha_1 \cdots \alpha_r}{}_{\sigma\beta_2 \cdots \beta_s} - \ldots -
       R^{\sigma}{}_{\beta_s\delta\gamma} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1}\sigma}
\end{align}</math>

This formula also applies to [[tensor density|tensor densities]] without alteration, because for the Levi-Civita (''not generic'') connection one gets:<ref name="lovelockrund" />
: <math>\nabla_{\mu}\left(\sqrt{g}\right) \equiv \left(\sqrt{g}\right)_{;\mu} = 0,</math> 
where 
: <math>g = \left|\det\left(g_{\mu\nu}\right)\right|.</math>

It is sometimes convenient to also define the purely covariant version of the curvature tensor by
:<math>R_{\sigma\mu\nu\rho} = g_{\rho\zeta} R^{\zeta}{}_{\sigma\mu\nu}.</math>