Part 1
Chapter 4 Definition: Riemannian Curvature Tensor
Definition Once a manifold is equipped with a connection, we can measure its curvature. The Riemannian curvature tensor (or Riemann tensor), denoted R, is the central tool used to quantify the curvature of a Riemannian manifold. It measures the extent to which the metric locally deviates from being flat.For any smooth vector fields X, Y, and Z, the curvature tensor R(X,Y)Z is a vector field defined by the failure of the second covariant derivatives to commute:
\[R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z\]Geometric Intuition
The curvature tensor reveals the effect of the manifold’s curvature on its geometry. Its most famous interpretation is that it measures the failure of a vector to return to its original state after being parallel transported around an infinitesimal closed loop (a parallelogram).
On a flat manifold (like a Euclidean plane), a vector transported around a closed loop will point in the same direction it started. On a curved manifold (like a sphere), it will be rotated. The curvature tensor quantifies this rotation.
A manifold is flat if and only if its Riemann curvature tensor is everywhere zero.
Curvature in Local Coordinates
In a local coordinate system, the components of the Riemann tensor, denoted $R^k_{lij}$, can be expressed entirely in terms of the Christoffel symbols ($\Gamma^k_{ij}$) of the Levi-Civita connection:
\[R^k\_{lij} = \\frac{\\partial \\Gamma^k\_{mj}}{\\partial x^l} - \\frac{\\partial \\Gamma^k\_{ml}}{\\partial x^j} + \\Gamma^p\_{mj} \\Gamma^k\_{pl} - \\Gamma^p\_{ml} \\Gamma^k\_{pj}\](Note: Some texts use different index conventions and signs, but this is a standard formulation).
This formula makes it clear that the curvature is determined entirely by the metric tensor $g$ and its derivatives, since the Christoffel symbols themselves are derived from $g$.