Editing Riemann curvature tensor (section)

=== Formally ===
When a vector in a Euclidean space is [[parallel transport]]ed around a loop, it will again point in the initial direction after returning to its original position. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a general [[Riemannian manifold]]. This failure is known as the non-[[holonomy]] of the manifold.

Let <math>x_t</math> be a curve in a Riemannian manifold <math>M</math>.  Denote by <math>\tau_{x_t}:T_{x_0}M \to T_{x_t}M</math> the parallel transport map along <math>x_t</math>.  The parallel transport maps are related to the [[covariant derivative]] by
: <math>
  \nabla_{\dot{x}_0} Y =
  \lim_{h\to 0} \frac{1}{h}\left(\tau^{-1}_{x_h}\left(Y_{x_h}\right) - Y_{x_0}\right) =
  \left.\frac{d}{dt}\left(\tau_{x_t}^{-1}(Y_{x_t})\right)\right|_{t=0}
</math>
for each [[vector field]] <math>Y</math> defined along the curve.

Suppose that <math>X</math> and <math>Y</math> are a pair of commuting vector fields.  Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of <math>x_0</math>.  Denote by <math>\tau_{tX}</math> and <math>\tau_{tY}</math>, respectively, the parallel transports along the flows of <math>X</math> and <math>Y</math> for time <math>t</math>.  Parallel transport of a vector <math>Z \in T_{x_0}M</math> around the quadrilateral with sides <math>tY</math>, <math>sX</math>, <math>-tY</math>, <math>-sX</math> is given by
: <math>\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z.</math>

The difference between this and <math>Z</math> measures the failure of parallel transport to return <math>Z</math> to its original position in the tangent space <math>T_{x_0}M</math>.  Shrinking the loop by sending <math>s, t \to 0</math> gives the infinitesimal description of this deviation:
: <math>\left.\frac{d}{ds}\frac{d}{dt}\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z\right|_{s=t=0} = \left(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}\right)Z = R(X, Y)Z</math>
where <math>R</math> is the Riemann curvature tensor.