Lecture Notes on General Relativity, Chapter 3: Curvature
by Sean M. Carroll · Caltech / arXiv gr-qc/9712019
The curvature chapter of Sean Carroll's graduate lecture notes. It covers parallel transport, derives the geodesic equation from both parallel transport and proper-time extremization, and treats affine parameters, the exponential map and geodesic deviation.
More resources on Geodesics
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
MathWorld: Geodesic
Wolfram MathWorld's reference entry on geodesics, giving the variational definition, the geodesic equation in terms of Christoffel symbols, and worked cases such as the sphere and plane. Useful for checking formulas and notation once you know basic differential geometry.
Lectures on Geometrical Anatomy of Theoretical Physics
Learn geometrical anatomy of theoretical physics with Frederic Schuller! Explore geodesics and more in this insightful course.
Tensor Calculus 15: Geodesics and Christoffel Symbols (extrinsic geometry)
The most approachable correct entry point: builds intuition from surfaces embedded in 3D before going intrinsic. A 21-minute visual derivation of the geodesic equation for a surface sitting in 3D space, explaining where Christoffel symbols come from. The next video in the playlist (part 16) works through geodesics on the plane and the sphere.
MIT 8.962 General Relativity, Lecture 9: Geodesics
A full graduate lecture from MIT's Spring 2020 general relativity course. It covers free fall as geodesic motion, curves that parallel transport their own tangent and extremize proper time, and quantities conserved along geodesics through spacetime symmetries.