Geodesics are curves whose velocity is parallel along itself. They are the straightest curves defined by the connection and, locally, the shortest curves defined by the metric.
Geodesic equation
Definition. A curve \(\gamma\) is a geodesic if \[ \frac{D\dot\gamma}{\mathrm{d}t}=0. \] In coordinates, \[ \ddot\gamma^k+\Gamma^k_{ij}(\gamma)\dot\gamma^i\dot\gamma^j=0. \]
Proposition. For each \((p,v)\in TM\), there is a unique maximal geodesic with \(\gamma(0)=p\), \(\dot\gamma(0)=v\). Its speed is constant.
Proof. The coordinate equation is a smooth first-order ODE on \(TM\), giving existence and uniqueness. Metric compatibility gives \[ \frac{\mathrm{d}}{\mathrm{d}t}|\dot\gamma|^2 =2g(D\dot\gamma/\mathrm{d}t,\dot\gamma)=0. \]
The exponential map and normal coordinates
Definition. For vectors for which the geodesic exists to time \(1\), \[ \exp_p(v)=\gamma_v(1). \]
Theorem. \(\mathrm{d}(\exp_p)_0=\operatorname{id}_{T_pM}\). Hence \(\exp_p\) is a diffeomorphism from a neighborhood of \(0\) onto a normal neighborhood of \(p\).
Proof. Scaling the geodesic parameter gives \(\exp_p(tv)=\gamma_v(t)\). Therefore \[ \mathrm{d}(\exp_p)_0v =\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{0}\exp_p(tv) =\dot\gamma_v(0)=v. \] The inverse function theorem supplies the local diffeomorphism.
Normal coordinates centered at \(p\) satisfy \[ g_{ij}(p)=\delta_{ij},\qquad \Gamma^k_{ij}(p)=0,\qquad \partial_kg_{ij}(p)=0. \] Curvature remains visible in second derivatives; normal coordinates remove coordinate acceleration at one point, not intrinsic curvature.
First variation and local minimization
Theorem (first variation of energy). For a variation \(\Gamma(s,t)\) of \(\gamma\), with variational field \(V\), \[ \left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_0E(\Gamma_s) =g(V,\dot\gamma)\big|_a^b -\int_a^bg\!\left(V,\frac{D\dot\gamma}{\mathrm{d}t}\right)\mathrm{d}t. \] Thus fixed-endpoint critical points of energy are geodesics.
Proof. Differentiate under the integral, use metric compatibility and torsion-freeness to interchange the covariant \(s\)- and \(t\)-derivatives, then integrate by parts: \[ \int g(DV/\mathrm{d}t,\dot\gamma)\,\mathrm{d}t =g(V,\dot\gamma)\big|_a^b-\int g(V,D\dot\gamma/\mathrm{d}t)\,\mathrm{d}t. \]
Gauss lemma. Radial geodesics are orthogonal to the images under \(\exp_p\) of spheres in \(T_pM\). Consequently the radial geodesic in a normal neighborhood uniquely minimizes length from \(p\).
Completeness
Theorem (Hopf-Rinow). For a connected Riemannian manifold, the following are equivalent: metric completeness; geodesic completeness; compactness of closed bounded sets. Under these conditions every two points are joined by a minimizing geodesic.
Example. The punctured plane with its Euclidean metric is incomplete: the curve \(t\mapsto(t,0)\), \(t\downarrow0\), has finite length but no endpoint in the manifold. Completeness is therefore global and is not implied by smoothness of the metric.