Geodesic equations and exponential maps

An affinely parameterized geodesic satisfies $D\dot\gamma/dt=0$. Expanding the covariant derivative gives the coordinate equation below. Smooth Christoffel coefficients turn this into a smooth first-order system for position and velocity, so standard ODE existence and uniqueness applies locally to every initial tangent vector. The parameter is part of the equation: affine changes of time preserve geodesic form, whereas an arbitrary nonlinear reparameterization can introduce tangential acceleration.

Metric compatibility makes the squared speed constant along a geodesic. The first variation of energy shows that fixed-endpoint critical curves are precisely these affinely parameterized geodesics. Being a critical point does not itself mean a global minimum. Sufficiently short geodesic segments minimize distance locally, but continuation past a cut point can lose minimality even though the same smooth geodesic equation continues to hold. Global route selection therefore requires information beyond local ODE uniqueness.

For $v\in T_pM$, let $\gamma_v$ be the geodesic with initial position p and initial velocity v. When it exists up to time one, define $\exp_p(v)=\gamma_v(1)$. Its domain contains an open neighborhood of the zero vector and is star shaped under scaling toward zero. The zero vector maps to p. Smooth dependence on ODE initial data makes the exponential smooth, and its derivative at zero is the identity after identifying the relevant tangent spaces naturally.

The inverse function theorem therefore provides a neighborhood on which the exponential is a diffeomorphism, giving normal coordinates near p. Radial lines in this tangent-space neighborhood map to geodesics. The radius of useful injectivity can be limited by singular derivatives or competing routes. Completeness will make the exponential defined on the entire tangent space, but it will not in general make it globally injective. A sphere is the clearest example of these distinct statements.

On the unit sphere, geodesics with nonzero speed are great circles. Every initial direction of length π reaches the antipodal point, so the exponential at p is far from injective there. Great circles can then continue indefinitely and return to previously visited points. This does not violate the initial-value uniqueness theorem, which fixes the entire initial velocity rather than only the endpoint. Different initial velocities can lead to the same endpoint.

The geodesic equation also must be interpreted on the correct manifold. A circle drawn in the Euclidean plane has nonzero intrinsic acceleration as a plane curve. The same circular motion, regarded as motion on the one-dimensional circle with its induced metric, is a geodesic when its angular speed is constant. Tangent spaces and the connection have changed. An extrinsic picture of a bent curve is therefore insufficient to decide whether its intrinsic covariant acceleration vanishes.