Geodesics

GPS satellites compute trajectories as geodesics in the Schwarzschild metric: GR correction +45 microseconds per day. Without it, GPS error grows by 11 km per day.

• **General Relativity**: planetary and photon trajectories are geodesics in the Schwarzschild metric. The bending of light around the Sun was predicted this way.
• **Riemannian ML**: gradient descent on hyperbolic space via exp_p and log_p is the foundation of Poincare embeddings.
• **Robotics**: optimal manipulator trajectories are geodesics in configuration space with an inertia metric.
• **Computer vision**: comparing shapes via geodesic distance on shape spaces underpins medical segmentation and motion capture.

Предварительные знания

• Riemannian metric tensor g_{ij} and covariant derivative
• Christoffel symbols and Levi-Civita connection
• Existence and uniqueness theorem for ODEs

• Riemann curvature
• Connection and parallel transport

The geodesic equation

GPS satellites compute ballistic trajectories as geodesics in the Schwarzschild metric: the GR correction is 45 microseconds per day. Without that correction GPS error would build up at 11 km per day.

Exponential map

In Riemannian-geometry-based deep learning (Poincare embeddings, Facebook AI 2017) the gradient-descent step is implemented through the exponential map on hyperbolic space. This improves the representation of hierarchical data by a factor of 10 over the Euclidean case.

Hopf-Rinow theorem and the cut locus

Hopf-Rinow theorem: a Riemannian manifold is geodesically complete if and only if it is complete as a metric space (Hopf-Rinow, 1931). The cut locus of a geodesic is the set of points up to which the geodesic stays length-minimizing, but beyond which it ceases to be.

Key ideas

• **Geodesic equation:** gamma''_k + Gamma^k_{ij} * gamma'_i * gamma'_j = 0. Christoffel symbols come from first derivatives of the metric.
• **Variational form:** geodesics are critical curves of L[gamma] = integral sqrt(g_{ij} * gamma'_i * gamma'_j) dt.
• **Exponential map:** exp_p(v) = gamma_v(1). Normal coordinates: g_{ij}(p) = delta_{ij}, Gamma^k_{ij}(p) = 0.