Riemann Curvature Tensor

Differential Geometry · Axiom Academy

LESSON The Riemann Curvature Tensor Understanding how manifolds curve through parallel transport, tensor components, and fundamental symmetries 1. Motivation: Path-Dependence of Parallel Transport On a curved manifold, parallel transporting a vector around a closed loop generally returns a different vector. This path-dependence is the signature of curvature. Consider a vector on a sphere: transport it along the equator, then up to the north pole, then back down. The final vector differs from the initial vector by a rotation. On a flat plane, this cannot happen—parallel transport is path-independent. 2. Definition: The Riemann Curvature Tensor The Riemann curvature tensor R(X,Y)Z is defined as the commutator of covariant derivatives minus the covariant derivative of the Lie bracket: Here X , Y , Z are vector fields, ∇ is the Levi-Civita connection, and [X,Y] is the Lie bracket. This measures the failure of ∇ X and ∇ Y to commute. In local coordinates (x 1 , ..., x n ), we write the components of the Riemann tensor as R l ijk , where: The explicit formula in terms of Christoffel symbols Γ is: This shows R measures how Christoffel symbols fail to be constant and how they fail to commute. 4. Symmetries and the First Bianchi Identity The Riemann tensor satisfies several crucial symmetries. First, lowering the upper index with the metric g lm , we get R ijkl = g lm R m ijk . This fully covariant tensor satisfies: Antisymmetry in first two indices: R ijkl = -R jikl

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