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 xt be a curve in a Riemannian manifold M. Denote by τxt : Tx0M → TxtM the parallel transport map along xt. 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 τtX and τtY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ Tx0M 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 Tx0M. Shrinking the loop by sending s, t → 0 gives the infinitesimal description of this deviation:
where R is the Riemann curvature tensor.
Read more about this topic: Riemann Curvature Tensor
Famous quotes containing the word meaning:
“A route differs from a road not only because it is solely intended for vehicles, but also because it is merely a line that connects one point with another. A route has no meaning in itself; its meaning derives entirely from the two points that it connects. A road is a tribute to space. Every stretch of road has meaning in itself and invites us to stop. A route is the triumphant devaluation of space, which thanks to it has been reduced to a mere obstacle to human movement and a waste of time.”
—Milan Kundera (b. 1929)