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Parallel Transport

This page documents practical triangle-frame transport and discrete connection tools.

Mathematical object

Transport is built from local orthonormal tangent frames:

  • per-face frame (t1, t2),
  • across-face transport as a 2×2 rotation aligning shared-edge tangent direction,
  • along-path transport by matrix composition.

Connection angle and loop holonomy are derived from those rotations.

Cochain space / degree

  • Tangent vectors are represented in local 2D frame coordinates (SVector{2}), or ambient tangential vectors (SVector{3}) for vertex transport interfaces.
  • Connection/holonomy are scalar angles (radians).

Orientation convention

  • Face orientation follows mesh orientation from SurfaceMesh.faces.
  • Shared-edge direction uses canonical primal edge orientation (i<j) for reproducibility.
  • Holonomy sign depends on loop traversal order.

Storage convention

  • Frames are stored as SMatrix{3,2} with columns (t1, t2).
  • transport_matrix_across_edge maps coordinates from faceL frame to faceR frame.

API

face_tangent_frames
vertex_tangent_frames
connection_angle_across_edge
transport_matrix_across_edge
parallel_transport_face_vector
parallel_transport_along_face_path
parallel_transport_vertex_vector
transport_edge_1form
rotate_in_tangent_frame
holonomy_along_cycle

Minimal example

using FrontIntrinsicOps
using LinearAlgebra

mesh = generate_icosphere(1.0, 1)
geom = compute_geometry(mesh)
topo = build_topology(mesh)

edgeid = findfirst(ef -> length(ef) == 2, topo.edge_faces)
fL, fR = topo.edge_faces[edgeid]
R = transport_matrix_across_edge(mesh, geom, fL, fR, edgeid)

v = R[:, 1]
v2 = parallel_transport_face_vector(v, mesh, geom, fL, fR, edgeid)

(orthogonality=norm(R' * R - I), norm_change=abs(norm(v2) - norm(v)))

Limitations and non-goals

  • Low-order frame-based transport only; no high-order connection discretization.
  • Vertex-to-vertex transport currently uses a simple dual-face path selector (graph/geodesic keyword maps to deterministic face-path traversal).
  • The current layer targets intrinsic research operators, not exact smooth Levi-Civita reconstruction.