Mathematical Foundations II: Curvature and Dynamics

Define an affine connection and the covariant derivative of a tensor field.

Derive the Levi-Civita connection from the metric compatibility and torsion-free conditions.

Define parallel transport along a curve and its relation to the covariant derivative.

Derive the geodesic equation as the curve of zero acceleration.

Define the Riemann curvature tensor in terms of the commutator of covariant derivatives.

Derive the algebraic symmetries of the Riemann tensor and the Bianchi identities.

Perform the Ricci decomposition of the Riemann tensor into the Ricci tensor, Ricci scalar, and Weyl tensor.

Interpret curvature physically via geodesic deviation.