Geodesics
Central Question
What replaces a straight line when straightness must be defined intrinsically?
Why This Matters
Geodesics generalize straight motion through a metric and a connection, exposing the distinction between local geometry and global shortest paths.
Learning Prompt
I know straight lines and variational ideas from mechanics. Develop geodesics on a Riemannian manifold by asking what it means to travel straight without an ambient reference. Compare zero covariant acceleration with stationarity of the energy functional and relate these to locally shortest paths. Work through the plane, sphere and cylinder. Make me distinguish a geodesic from a globally minimizing path, and separate parametrization from the geometric curve. Explain where a metric and a connection enter, then connect geodesic motion with mechanics and the role of spacetime geometry in relativity.