Shortest path (computed using the Metric tensor). wiki

Solve using Calculus of variations:

1. Definition of a family of smooth curves

The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by

see equations here for a 2D Surface

intro

can compute wavefront using Eikonal equation

A curve is a geodesic iff its acceleration is normal to the surface S at every point, that is its velocity doesn't change in the tangent plane, at every point! (it's locally straight when projected to the surface).

You can think of geodesics as the trajectories that a little frictionless ball would follow in a "hollow" surface, where the only forces acting on it are the Normal forces due to the walls of the hollow surface

This means that the Geodesic curvature of a geodesic is $0$, which makes sense, as the geodesic is supposed to not be accelerating in the realm of the surface

Video

Geodesics are preserved under Isometryes

Solution to geodesic given point and tangent vector exists and is unique, at least locally, and it also depends smoothly on these initial conditions.

Examples

Geodesics on the plane, are Lines

Geodesics on Surface of revolutions. Meridians are geodesics. Parallel (surface of revolution) are geodesics when they intersect a meridian at a critical point (vertically)

Geodesics on the Sphere are Maximal circles

Geodesic coordinates