Geodesic Calculus & Riemannian Propagation

• Geodesic calculus is a coordinate‑independent framework for defining and computing geodesics, exponential/logarithm maps, and parallel transport on Riemannian and sub‑Riemannian manifolds.
• Riemannian propagation employs analytic and discrete methods to transport tangent-space structures, enabling interpolation, extrapolation, and optimization in manifold-valued data.
• The framework unifies Hamiltonian, variational, and discrete approaches, offering robust tools for curvature analysis, statistical propagation, and algorithmic design in geometric settings.

Geodesic calculus and Riemannian propagation form the analytic backbone of both classical and modern differential geometry. Geodesic calculus provides a rigorous, coordinate-independent framework to define, characterize, and compute geodesics, exponential maps, logarithm maps, and parallel transport on Riemannian and sub-Riemannian manifolds. Riemannian propagation encompasses the suite of algorithms and analytic techniques used to transport, interpolate, and extrapolate tangent-space and higher-order structures along geodesics, with powerful applications in optimization, geometric analysis, control, and the study of manifold-valued data.

1. Foundations: Riemannian and Sub-Riemannian Geodesic Theory

A Riemannian manifold $(M,g)$ provides a metric $g\in \Gamma(Sym^2 T^*M)$, inducing an inner product on each tangent space $T_xM$. A geodesic is a curve $\gamma: I \to M$ whose tangent vector is parallel transported along itself, i.e., $\nabla_{\dot\gamma}\dot\gamma=0$. In local coordinates and with Christoffel symbols $\Gamma^k_{ij}$, the geodesic equation is

$\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$

defining the geodesic flow as the solution flow of these ODEs (Ghojogh, 4 May 2026, Opara, 2017).

Sub-Riemannian geometry generalizes this structure by introducing a horizontal distribution $H\subset TM$ with an inner product $h$ only defined on $H$, leading to normal sub-Riemannian geodesics as projections of Hamiltonian flows for the Hamiltonian $g\in \Gamma(Sym^2 T^*M)$0, where $g\in \Gamma(Sym^2 T^*M)$1 is the degenerate cometric vanishing on $g\in \Gamma(Sym^2 T^*M)$2. When $g\in \Gamma(Sym^2 T^*M)$3 admits a "taming" Riemannian metric $g\in \Gamma(Sym^2 T^*M)$4 with $g\in \Gamma(Sym^2 T^*M)$5, the geodesic flows of $g\in \Gamma(Sym^2 T^*M)$6 and $g\in \Gamma(Sym^2 T^*M)$7 interact in a highly structured way (Molina et al., 2015).

The key technical object for the interaction of these flows is the connection $g\in \Gamma(Sym^2 T^*M)$8 on $g\in \Gamma(Sym^2 T^*M)$9 defined via projections onto horizontal and vertical subbundles (see Section 2 below), and the commutation of the geodesic flows is characterized via the $T_xM$0-parallelism of $T_xM$1.

2. Hamiltonian, Variational, and Discrete Geodesic Calculus

The geodesic calculus admits several analytic and computational formulations:

• Hamiltonian Formalism: The Hamiltonian vector fields for $T_xM$2 and $T_xM$3 generate the (sub-)Riemannian flows on $T_xM$4 with canonical symplectic structure, and their commutation is controlled by the parallelism condition $T_xM$5 (Molina et al., 2015, Wang, 2020).
• Variational Formulation: Geodesics are critical points (often minimizers, locally) of the energy functional

$T_xM$6

leading via the Euler–Lagrange equations to precisely the same ODE as above. Small variations yield the Jacobi equation for infinitesimal spread of geodesics (Opara, 2017).

• Discrete Geodesic Calculus: Variational discretization replaces the continuous energy by a sum (or product integral) over local increments:

$T_xM$7

with $T_xM$8 to consistent order. Discrete logarithm and exponential maps are given by first and higher increments, converging strongly to the continuous objects as $T_xM$9 (Rumpf et al., 2012, Rumpf et al., 2012, Beutler et al., 15 May 2025, Hartwig et al., 10 Oct 2025). Schild's ladder and similar parallelogram constructions yield discrete parallel transport with rigorous $\gamma: I \to M$0 convergence.

• Riemannian Flow Bundle Approaches: The geodesic flow bundle (GFB) formalism bypasses metric computation by directly assembling all arclength-parameterized geodesics in terms of sphere/triangle constructions, using the curvature $\gamma: I \to M$1 as generating data, and integrating via power series and product integrals (Boitier et al., 2022).

3. Exponential, Logarithm Maps, and Propagation Operators

At the analytic core are exponential and logarithm maps:

• Exponential map $\gamma: I \to M$2: Returns the point reached after unit arclength traveling in direction $\gamma: I \to M$3 along the geodesic. Locally, this map admits second- and higher-order expansions in coordinates:

$\gamma: I \to M$4

for small $\gamma: I \to M$5 (Ghojogh, 4 May 2026, Rumpf et al., 2012, Hartwig et al., 10 Oct 2025).

• Logarithm map $\gamma: I \to M$6: The inverse, returning the initial velocity $\gamma: I \to M$7 such that $\gamma: I \to M$8.
• Parallel transport $\gamma: I \to M$9: Describes the unique (in simply connected, geodesically complete settings) linear isometry from $\nabla_{\dot\gamma}\dot\gamma=0$0 to $\nabla_{\dot\gamma}\dot\gamma=0$1 along a geodesic, solving

$\nabla_{\dot\gamma}\dot\gamma=0$2

for a vector field $\nabla_{\dot\gamma}\dot\gamma=0$3 along $\nabla_{\dot\gamma}\dot\gamma=0$4.

• Jacobi Fields: Linearization of the geodesic flow produces Jacobi fields $\nabla_{\dot\gamma}\dot\gamma=0$5 encoding the divergence and convergence of nearby geodesics, crucial for curvature computations and propagation estimates via the Jacobi equation $\nabla_{\dot\gamma}\dot\gamma=0$6 (Opara, 2017).

For matrix manifolds, especially SPD and Stiefel manifolds, explicit closed forms for exponential, logarithm, and parallel transport are available and essential for Riemannian optimization (Ghojogh, 4 May 2026).

4. Commutation, Submersion, and Horizontal Lifting

When a sub-Riemannian structure $\nabla_{\dot\gamma}\dot\gamma=0$7 is tamed by a Riemannian metric $\nabla_{\dot\gamma}\dot\gamma=0$8 with $\nabla_{\dot\gamma}\dot\gamma=0$9, and when the vertical bundle $\Gamma^k_{ij}$0 fibers are totally geodesic, the flows for $\Gamma^k_{ij}$1 and $\Gamma^k_{ij}$2 commute if and only if $\Gamma^k_{ij}$3. In this scenario:

• Sub-Riemannian geodesics are precisely the horizontal lifts of Riemannian geodesics in the base manifold $\Gamma^k_{ij}$4 under a Riemannian submersion $\Gamma^k_{ij}$5,
• The sub-Riemannian exponential map is computed by first applying the Riemannian exponential in $\Gamma^k_{ij}$6, then performing a vertical correction via Levi–Civita parallel transport,
• In principal bundle cases (e.g., Yang–Mills charged particle dynamics), explicit formulas involve left/right group actions and parallel transport in the group (Molina et al., 2015).

This provides a unified framework for understanding Riemannian and sub-Riemannian propagation as well as geometric control.

5. Discrete and Algorithmic Propagation: Shapes, Data, and Latent Spaces

Time-discrete geodesic calculus generalizes to finite- and infinite-dimensional settings (e.g., shape space, Sobolev curve manifolds), with provable convergence via $\Gamma^k_{ij}$7-convergence and error estimates (Beutler et al., 15 May 2025, Rumpf et al., 2012). The discrete exponential and logarithm maps, together with variational or Schild-style parallel transport, generalize Riemannian propagation to spaces of shapes, curves, or even implicit manifolds in machine learning latent spaces (Hartwig et al., 10 Oct 2025). The essential principles are:

• Discrete geodesics are constructed as energy-minimizers in a product geometry,
• Discrete log/exponential maps converge strongly to the continuous limits,
• Discrete parallel transport\—via Schild's ladder/parallelogram or intrinsic Euler–Lagrange steps—preserves tangent-covariant data and enables geometric interpolation/extrapolation,
• All operators have proven error rates, e.g., $\Gamma^k_{ij}$8 for geodesics and parallel transport, $\Gamma^k_{ij}$9 for one-sided finite difference covariant derivatives (Beutler et al., 15 May 2025).

A robust computational pipeline has emerged for applying geodesic calculus in data science settings—e.g., learning a projection map $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$0 to latent data manifolds and equipping the induced submanifold with a Riemannian metric via the pullback of the ambient geometry. Christoffel symbols and connection coefficients are then assembled via automatic differentiation of $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$1, and geodesic shooting or discrete variational principles solve for shortest paths in data (Hartwig et al., 10 Oct 2025).

6. Riemannian Propagation of Tensors and Statistical Quantities

Riemannian propagation schemes extend to gradients, Hessians, and covariance structures. The Riemannian gradient at $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$2 is defined via $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$3, while the Hessian is the covariant derivative of the gradient:

$\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$4

with explicit coordinate expressions involving Christoffel symbols.

Gaussian distributions specified in tangent spaces at $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$5 propagate along geodesics via pushforward by exponential map and covariance matrices evolve by parallel transport (Ghojogh, 4 May 2026). These techniques yield manifold-valued stochastic processes essential for modeling in geometric statistics and manifold learning.

In practical algorithms (optimization, processing of geometric data), retractions are standard first-order surrogates for the exponential map, balancing efficiency and geometric accuracy (Ghojogh, 4 May 2026).

7. Propagation Beyond Riemannian Geometry and Structural Hamiltonian Perspectives

Geodesic calculus admits extension to pseudo-Riemannian geometries via geodesic flow bundle or GFB formalism, where all arclength-parametrized geodesics are generated from curvature data via infinitesimal spherical (or hyperbolic) triangles and product-integral constructions (Boitier et al., 2022). This framework directly relates the variation of geodesics to Gaussian curvature and, via expansion, gives closed-form higher-order corrections.

The generalized covariant Hamilton system (GCHS) built from the generalized structural Poisson bracket (GSPB) endows $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$6 dynamics with explicit dependence on the Christoffel symbols and their traces (S-dynamics, geospin matrices), recovering the geodesic equation from covariant equilibrium in this Hamiltonian setting (Wang, 2020). This demonstrates the deep interplay between geometric structures (connection coefficients) and Hamiltonian system theory, and emphasizes the manifestly covariant, gauge-independent approach to geodesic propagation.

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Summary Table: Core Operators of Geodesic Calculus and Propagation

Operator / Construction	Continuous Manifold Version	Discrete / Computational Version
Geodesic equation	$\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$7 (ODE with $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$8)	Variational time discrete minimizers, e.g. $\ddot x^k + \Gamma^k_{ij}(x)\dot x^i\dot x^j=0,$9
Exponential map $H\subset TM$0	Solution at $H\subset TM$1 of geodesic IVP	Stepwise discrete shooting, $H\subset TM$2
Logarithm map $H\subset TM$3	Initial velocity of geodesic from $H\subset TM$4 to $H\subset TM$5	First step increment, $H\subset TM$6
Parallel transport	ODE for vector fields, or Jacobi fields	Schild's ladder, discrete parallelogram construction
Curvature / Jacobi analysis	Jacobi ODE, $H\subset TM$7	Finite difference or variational discrete curvature tensors

This framework—combining analytic, variational, discrete, and algorithmic methods—provides a unified foundation for the computation, analysis, and deployment of geodesic flows and Riemannian propagation across geometry, analysis, and data science (Molina et al., 2015, Ghojogh, 4 May 2026, Opara, 2017, Rumpf et al., 2012, Beutler et al., 15 May 2025, Wang, 2020, Boitier et al., 2022, Hartwig et al., 10 Oct 2025, Rumpf et al., 2012).