Geodesics and parallel transport are the dynamical objects determined by a connection. They describe straightness, not necessarily shortest distance unless the connection is Levi-Civita.
Geodesic equation
A geodesic has zero covariant acceleration.
Definition. A curve \(\theta(t)\) is a \(\nabla\)-geodesic if \[ \nabla_{\dot\theta}\dot\theta=0. \] In coordinates, \[ \ddot\theta^k+\Gamma^k_{ij}(\theta)\dot\theta^i\dot\theta^j=0. \]
Proposition. For a flat connection with affine coordinates, geodesics are straight lines in those coordinates.
Proof. In affine coordinates for a flat connection, \(\Gamma^k_{ij}=0\). The geodesic equation becomes \(\ddot\theta^k=0\), whose solutions are \(\theta^k(t)=a^k+tb^k\).
Parallel transport
Parallel transport carries tangent vectors along a curve while keeping covariant derivative zero.
Definition. A vector field \(V(t)\) along \(\theta(t)\) is parallel if \[ \nabla_{\dot\theta}V=0. \] In coordinates, \(\dot V^k+\Gamma^k_{ij}\dot\theta^iV^j=0\).
Remark. For the Levi-Civita connection, parallel transport preserves inner products. For general statistical connections, this preservation may hold only in dual pairing with the conjugate connection.
Levi-Civita geodesics
For the metric connection, geodesics are locally length minimizing.
Theorem. The Levi-Civita connection is the unique torsion-free connection satisfying \(\nabla g=0\). Its geodesics solve the Euler-Lagrange equations of the energy functional \[ E(\theta)=\frac12\int g_{ij}\dot\theta^i\dot\theta^j\,\mathrm{d}t. \]
Proof. Metric compatibility and torsion-freeness determine the Christoffel symbols by the Koszul formula. Varying the energy functional and integrating by parts yields the geodesic equation with these symbols.
Statistical straightness
Mixture and exponential geodesics can be straight without minimizing Fisher-Rao length.
Remark. This is why information geometry distinguishes affine geodesics from Riemannian distance geodesics. The former encode statistical linearity; the latter encode Fisher length.
Autoparallels and the geodesic equation
Definition. A curve \(\gamma(t)\) is an affinely parametrized \(\nabla\)-geodesic when \(\nabla_{\dot\gamma}\dot\gamma=0\). In coordinates, \[ \ddot\theta^k+\Gamma^k{}_{ij}(\theta)\dot\theta^i\dot\theta^j=0. \]
For a general affine connection, this means its velocity transports itself. For the Levi--Civita connection, the same curves are locally stationary points of energy and locally minimize length when restricted before their first conjugate point.
Proposition. Along a Levi--Civita geodesic, speed \(g(\dot\gamma,\dot\gamma)\) is constant.
Proof. Metric compatibility and the geodesic equation give \[ \frac{\mathrm{d}}{\mathrm{d}t}g(\dot\gamma,\dot\gamma) =2g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)=0. \]
Parallel transport and holonomy
A vector field \(V(t)\in T_{\gamma(t)}\mathcal M\) is parallel when \[ \nabla_{\dot\gamma}V=0,\qquad \dot V^k+\Gamma^k{}_{ij}\dot\theta^iV^j=0. \] This linear ODE determines an isomorphism \(\mathcal P_\gamma:T_{\gamma(0)}\mathcal M\to T_{\gamma(1)}\mathcal M\). If \(\nabla\) is metric compatible, parallel transport preserves inner products. Curvature is the infinitesimal failure of transport around a loop to return a vector unchanged.
Exponential maps and normal coordinates
For \(v\in T_p\mathcal M\), let \(\gamma_v\) be the geodesic with \(\gamma_v(0)=p\), \(\dot\gamma_v(0)=v\). The affine exponential map is \(\operatorname{Exp}_p^\nabla(v)=\gamma_v(1)\), whenever defined. For the Levi--Civita connection, sufficiently small \(v\) provide normal coordinates, in which \(\Gamma^k{}_{ij}(p)=0\) and radial geodesics are straight rays. Metric first derivatives vanish at \(p\), but curvature remains in second-order terms.
Example (three notions of straightness). On a categorical simplex, mixture segments are geodesics of the mixture connection, normalized geometric interpolations \(p_i(t)\propto p_i(0)^{1-t}p_i(1)^t\) are geodesics of the exponential connection, and square-root great circles are geodesics of the Levi--Civita connection. They coincide only in exceptional one-dimensional or symmetric configurations.
Jacobi fields and statistical sensitivity
A variation through Levi--Civita geodesics has variation field \(J\) satisfying \[ \nabla_{\dot\gamma}^2J+R(J,\dot\gamma)\dot\gamma=0. \] Jacobi fields quantify how nearby geodesics separate. Positive curvature tends to focus geodesics, negative curvature to disperse them. In statistical manifolds this controls sensitivity of geodesic interpolation and uniqueness of intrinsic means, though inference itself is usually governed first by local Fisher geometry rather than global distance.