Knowlpedia

Definition

Pullback connection on a vector bundle

The canonical connection on a pullback vector bundle induced by a connection on the original bundle.

Definition

Let f:N→Mf:N\to M be a , let E→ME\to M be a smooth real or complex , and let ∇\nabla be a on EE. The pullback connection f∗∇f^*\nabla is the unique connection on the f∗E→Nf^*E\to N such that

(f∗∇)(f∗s)=f∗(∇s)(f^*\nabla)(f^*s)=f^*(\nabla s)

for every local ss of EE. Here the right side is the pullback of the EE-valued 11-form ∇s\nabla s, so it is a on NN.

Local construction

Every local section of f∗Ef^*E can be written as a finite sum σ=∑iaif∗si\sigma=\sum_i a_i f^*s_i. The Leibniz rule forces

(f∗∇)σ=∑idai⊗f∗si+∑iaif∗(∇si).(f^*\nabla)\sigma = \sum_i da_i\otimes f^*s_i +\sum_i a_i f^*(\nabla s_i).

This formula is independent of the chosen expression for σ\sigma. In a local frame with ∇=d+A\nabla=d+A, the pullback connection is d+f∗Ad+f^*A. The construction and its frame independence are treated in Tu, chapters on connections and pullbacks.

Curvature and functoriality

is natural under pullback:

Rf∗∇=f∗R∇.R^{f^*\nabla}=f^*R^\nabla.

Thus a pullback of a flat connection is flat, although the converse need not hold because ff may miss directions on which the original curvature is nonzero. Pullback is functorial: for g:L→Ng:L\to N, the canonical identification (f∘g)∗E≅g∗f∗E(f\circ g)^*E\cong g^*f^*E carries (f∘g)∗∇(f\circ g)^*\nabla to g∗(f∗∇)g^*(f^*\nabla). These naturality statements follow from the local pullback formula; see Tu, §25.

If i:S↪Mi:S\hookrightarrow M is an , i∗∇i^*\nabla is the restriction of ∇\nabla to directions tangent to SS.

Examples and scope

For a constant map f:N→{x}⊂Mf:N\to\{x\}\subset M, the pullback bundle is canonically N×ExN\times E_x, and f∗∇f^*\nabla is the trivial connection in this identification. Pulling back the Levi–Civita connection along a curve gives the covariant derivative used to define parallel along that curve.

References
  1. Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: §25, pullback connections and curvature.
  2. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, mappings and induced connections.