Editing Riemann curvature tensor (section)

== Geometric meaning ==
[[File:Riemann_curvature_motivation_shpere.gif|thumbnail|Figure showing the geometric meaning of the Riemann curvature tensor in a spherical curved manifold. The fact that this transfer can define two different arrows at the starting point gives rise to the Riemann curvature tensor. The orthogonal symbol indicates that the [[dot product]] (provided by the metric tensor) between the transmitted arrows (or the tangent arrows on the curve) is zero. The angle between the two arrows is zero when the space is flat and greater than zero when the space is curved. The more curved the space, the greater the angle.]]

=== Informally ===
One can see the effects of curved space by comparing a tennis court and the Earth. Start at the lower right corner of the tennis court, with a racket held out towards north. Then while walking around the outline of the court, at each step make sure the tennis racket is maintained in the same orientation, parallel to its previous positions. Once the loop is complete the tennis racket will be parallel to its initial starting position. This is because tennis courts are built so the surface is flat. On the other hand, the surface of the Earth is curved: we can complete a loop on the surface of the Earth. Starting at the equator, point a tennis racket north along the surface of the Earth. Once again the tennis racket should always remain parallel to its previous position, using the local plane of the horizon as a reference. For this path, first walk to the north pole, then walk sideways (i.e. without turning), then down to the equator, and finally walk backwards to your starting position. Now the tennis racket will be pointing towards the west, even though when you began your journey it pointed north and you never turned your body. This process is akin to [[parallel transport]]ing a vector along the path and the difference identifies how lines which appear "straight" are only "straight" locally. Each time a loop is completed the tennis racket will be deflected further from its initial position by an amount depending on the distance and the curvature of the surface. It is possible to identify paths along a curved surface where parallel transport works as it does on flat space. These are the [[geodesics]] of the space, for example any segment of a great circle of a sphere.

The concept of a curved space in mathematics differs from conversational usage. For example, if the above process was completed on a cylinder one would find that it is not curved overall as the curvature around the cylinder cancels with the flatness along the cylinder, which is a consequence of [[Gaussian curvature]] and Gauss's [[Theorema Egregium]]. A familiar example of this is a floppy pizza slice, which will remain rigid along its length if it is curved along its width.

The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved surface).

=== 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.