Editing Riemann curvature tensor (section)

=== Surfaces ===
For a two-dimensional [[surface (topology)|surface]], the Bianchi identities imply that the Riemann tensor has only one independent component, which means that the [[Ricci scalar]] completely determines the Riemann tensor. There is only one valid expression for the Riemann tensor which fits the required symmetries:
: <math>R_{abcd} = f(R) \left(g_{ac}g_{db} - g_{ad}g_{cb}\right)</math>
and by contracting with the metric twice we find the explicit form:
: <math>R_{abcd} = K\left(g_{ac}g_{db} - g_{ad}g_{cb}\right) ,</math>
where <math>g_{ab}</math> is the [[metric tensor]] and <math>K = R/2</math> is a function called the [[Gaussian curvature]] and <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> take values either 1 or&nbsp;2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the [[sectional curvature]] of the surface.  It is also exactly half the [[scalar curvature]] of the 2-manifold, while the [[Ricci curvature]] tensor of the surface is simply given by
: <math>R_{ab} = Kg_{ab}.</math>