Riemann Curvature Tensor - Geometrical Meaning

Geometrical Meaning

When a vector in a Euclidean space is parallel transported 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 holonomy of the manifold.

Let x_t be a curve in a Riemannian manifold M. Denote by τ_xt : T_x0M → T_xtM the parallel transport map along x_t. The parallel transport maps are related to the covariant derivative by

for each vector field Y defined along the curve.

Suppose that X and Y are a pair of commuting vector fields. Each of these fields generates a pair of one-parameter groups of diffeomorphisms in a neighborhood of x₀. Denote by τ_tX and τ_tY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ T_x0M around the quadrilateral with sides tY, sX, −tY, −sX is given by

This measures the failure of parallel transport to return Z to its original position in the tangent space T_x0M. Shrinking the loop by sending s, t → 0 gives the infinitesimal description of this deviation:

where R is the Riemann curvature tensor.