parallel transport

平行移动

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

Given a connection on a smooth vector bundle and a smooth path in its base, the linear map between endpoint fibers that sends an initial vector to the endpoint value of the unique section along the path with zero covariant derivative and that initial value.

给定光滑向量丛上的联络及底空间中的光滑道路，由指定初值且沿道路协变导数为零的唯一截面的终值，定义两端纤维之间的线性映射。

Example

For the Euclidean connection on $T\mathbb R^2$, a vector with constant Cartesian components remains parallel along any smooth path.

Usage and distinctions

Transport depends on a connection and generally on the whole path, not just its endpoints. A metric-compatible connection preserves inner products, but this is additional to being a connection.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§3.3.2, Parallel transport, initial-value problem (3.3.7) · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二：平行移动和协变微分；PDF p. 77, printed p. 74 · Checked 2026-10-14

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