---
title: Straightest Geodesics in Curved Spaces
url: https://www.emergentmind.com/topics/straightest-geodesics
type: topic
---

# Straightest Geodesics in Curved Spaces

A straightest geodesic is a curve on a manifold whose tangent vector is parallel transported along itself with respect to a given connection, providing a rigorous generalization of the concept of "straight lines" to curved or structured spaces. In contrast to shortest (metric) geodesics, which minimize length or energy functionals, straightest geodesics are defined purely through connection-based parallelism. In classical Riemannian geometry, these notions coincide: both are characterized as critical points of the energy or length and as solutions to the geodesic equation for the Levi-Civita connection. However, in the presence of nonholonomic constraints, non-metric connections, or generalized geometric settings (e.g., discrete, sub-Riemannian, teleparallel, or data-driven manifolds), the distinction becomes essential and operationally significant.

## 1. Definitions and Fundamental Equations

The straightest geodesic on a smooth manifold \((M, g)\) under a connection \(\nabla\) is a curve \(\gamma(\lambda)\) such that its tangent vector is autoparallel:
\[
\nabla_{\dot\gamma}\dot\gamma = 0
\]
In local coordinates, for a connection with Christoffel symbols \(\Gamma^{k}_{ij}\), the geodesic equations read:
\[
\frac{d^2x^k}{d\lambda^2} + \Gamma^k_{ij}\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda}=0
\]
For the Levi-Civita connection (the unique torsion-free, metric-compatible connection for \(g\)), these are also the locally length-minimizing paths (metric geodesics) [2305.15228]. In spaces with more general connections—such as in modified symmetric teleparallel gravity [2412.15805] with a torsionless, flat, but non-metric connection, or in sub-Riemannian geometries [1909.08275]—these equations must be formulated with care, and autoparallels may differ from metric geodesics.

## 2. Distinction Between Straightest and Shortest Geodesics

### Riemannian and Sub-Riemannian Geometries

In classical Riemannian geometry, the straightest and shortest concepts agree due to the equivalence of metric variation and parallel transport under the Levi-Civita connection [2305.15228]. In sub-Riemannian geometry, several notions diverge:
- **Straightest (Autoparallel) Geodesics:** Solutions to any of the partial connections adapted to the horizontal distribution \(D\) (such as the Schouten, d'Alembert, or Cartan-Morimoto connections), defined by
  \[
  \nabla^*_V V = 0
  \]
  with \(\nabla^*\) the chosen partial or Cartan connection [1909.08275].
- **Shortest (Variational) Geodesics:** Curves that are critical points of the length functional, i.e., satisfy the Euler-Lagrange equations restricted to admissible directions.

These notions coincide only under additional symmetry or integrability conditions, such as in homogeneous spaces, Chaplygin systems, or contact-Sasaki manifolds. Otherwise, straightest and shortest geodesics can be systematically distinct [1909.08275].

### Modified Gravity and Other Connections

In modified symmetric teleparallel gravity (often \(f(Q)\) gravity), the spacetime admits a flat, torsionless connection \(\Gamma\) independent of the Levi-Civita connection \(\mathring{\Gamma}\) for \(g_{\mu\nu}\) [2412.15805]. There are two natural geodesic equations:
- **Autoparallel (Straightest) Geodesics:**
  \[
  \frac{d^2 x^\lambda}{d \tau^2} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = 0
  \]
- **Metric (Shortest) Geodesics:**
  \[
  \frac{d^2 x^\lambda}{d \tau^2} + \mathring{\Gamma}^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = 0
  \]
Here, the field equations and the generalized Bianchi identities guarantee that physical free-fall follows metric geodesics, not autoparallels, except in the special case where the non-metricity scalar is constant and the connections coincide [2412.15805]. Thus, "straightest" and "shortest" are distinct, with empirical trajectories governed by the shortest path.

## 3. Discrete Straightest Geodesics

Discretized geometric domains, such as lattices, meshes, or cellular automata, require operational definitions of straightest geodesics not relying on metric minimization alone. Arrighi and Dowek [1507.06836] define straightest discrete geodesics via a deviation function:
\[
\Delta(E,F,G) = \sum_{\mu = 0}^n \left( \frac{l(E, F^{\mu-}, G) - l(E, F^{\mu+}, G)}{2} \right)^2
\]
where \(l(E,F,G)\) is the broken line length for three lattice points and \(F^{\mu\pm}\) are neighboring shifts. Discrete geodesics are sequences minimizing \(\Delta\) at each intermediate point. In the continuum limit, this construction converges to the autoparallel geodesic ODE, guaranteeing that the autoparallel principle is recovered for smooth metric spaces.

For triangular meshes and computer graphics contexts, straightest geodesics are constructed using angle-preservation at mesh vertices and are traced iteratively, preserving parallel transport rules under mesh geometry [2603.15780].

## 4. Straightest Geodesics in Structured and Data-Driven Geometries

### Deep Linear Networks
In the geometry of deep linear networks, straightest geodesics correspond to "horizontal" straight lines in a balanced product manifold of matrix factors, projected via Riemannian submersion to the space of full-rank matrices [2510.07324]. Explicit ODEs and closed-form solutions are obtained for geodesics under the induced metric, with affine lifts in the balanced manifold yielding the straightest projected geodesics in parameter space.

### Data-Driven and Differentiable Manifolds
Geodesics arising in the context of machine learning and differentiable manifolds are constructed via Hamiltonian dynamical systems:
\[
H(q,p) = \frac{1}{2} g^{ij}(q) p_i p_j
\]
with straightest geodesics given by solutions to the Hamiltonian flow with exact energy conservation. Distance fields and geodesic flows can be efficiently computed by solving the Eikonal equation with curvature-aware loss scaling, focusing learning resources on regions of strong geodesic deviation [2305.15228].

## 5. Straightest Geodesics in Sub-Riemannian and Carnot Structures

In sub-Riemannian jet spaces \(J^k\), straightest geodesics are constructed via polynomial data, with the structure of geodesics determined by properties of the polynomial \(F(x)\) that arises in a canonical normal form. Necessary and sufficient conditions for minimizers are provided: only constant \(F\) (horizontal lines) and certain "seagull" polynomials of high even degree, giving rise to direct heteroclinic connections, yield globally minimizing straightest geodesics [2109.13835]. The analysis leverages Hamilton-Jacobi methods, magnetic reductions, and period asymptotics to determine minimization and cut times.

Homogeneous sub-Riemannian geometries—including Carnot groups, contact manifolds, and symmetric flag manifolds—admit a coincidence between various notions of straightest geodesics (d'Alembert, Schouten, Cartan-Morimoto) and shortest (Hamiltonian or variational) geodesics, but this is a special property of their structure [1909.08275].

## 6. Physical and Computational Implications

- **Modified Theories of Gravity:** In \(f(Q)\) gravity, despite the existence of two inequivalent connections, physical free-fall follows the metric (shortest) geodesics, and the straightest geodesics of the teleparallel connection are not physical unless the theory reduces to GR [2412.15805].
- **Computational Geometry and Graphics:** Mesh-based straightest geodesics underpin robust algorithms for tracing morphological features, parallel-transport, and defining log/exp maps for Riemannian optimizers (e.g., LBFGS), with GPU-based differentiable implementations supporting large-scale learning tasks [2603.15780].
- **Sub-Riemannian Control:** The multiplicity of straightest geodesic definitions in nonholonomic mechanics reveals deep relationships between controllability, metric invariants, and geometric synthesis, with homogeneous models providing rare cases of full equivalence.

## 7. Example Table: Notions of “Straightest Geodesic” Across Geometric Contexts

| Context                         | Straightest Geodesic Definition                       | Coincidence with Shortest?      |
|----------------------------------|-------------------------------------------------------|---------------------------------|
| Riemannian manifold              | Levi-Civita autoparallel                             | Always                          |
| f(Q) gravity                     | Autoparallel of non-metric, torsion-free connection  | Only if f(Q) is linear/constant |
| Sub-Riemannian general           | Partial/Cartan/d'Alembert connections                | Rare; only in homogeneous cases |
| Deep linear networks             | Affine horizontal lift in balanced manifold          | For “aligned” endpoint case     |
| Discretized manifold/mesh        | Angle-preserving/parallel-transport discrete rule    | Only as discretization → smooth |

The equivalence or divergence of the “straightest” and “shortest” concepts is determined by the ambient geometry, choice of connection, and the structure of constraints. In computational and machine learning settings, operational definitions tailored to the discrete or data-driven manifold structure are essential, often leveraging symplectic integration and parallel computation for efficient geodesic tracing and differential learning. The nuanced distinction between these two notions remains a central theme in the study of geometry, control, and physical theories.

Source: https://www.emergentmind.com/topics/straightest-geodesics