流形间光滑映射保线性联络的定义问询

Awesome question—let's unpack what it means for a smooth map (f: M \to M') to preserve the linear connections (D) (on (M)) and (D') (on (M')). There are a few equivalent ways to frame this, depending on whether you prefer abstract coordinate-free language, geometric intuition, or concrete coordinate calculations:

Core Coordinate-Free Definition #

At its heart, preserving connections means that the pushforward of the map commutes with the covariant derivatives defined by each connection. Here's the precise statement:

• For every pair of smooth vector fields (X, Y) on (M), the pushforward of the covariant derivative of (Y) along (X) (using (D)) must equal the covariant derivative of the pushforward of (Y) along the pushforward of (X) (using (D')). In symbols:
  [
  f_* (D_X Y) = D'{f* X} (f_* Y)
  ]
  Think of this as: pushing forward after computing the covariant derivative is exactly the same as computing the covariant derivative after pushing forward. The map doesn't "distort" the way we differentiate vector fields along other vector fields.

Geometric Intuition via Geodesics #

A more visual, geometric way to characterize preserved connections is through geodesics:

• (f) preserves the connections if and only if it maps every geodesic of (M) (with respect to (D)) to a geodesic of (M') (with respect to (D')), and it preserves the affine parameterization of those geodesics.
  • Note: This isn't just mapping geodesics to geodesics—if (\gamma(t)) is a geodesic with parameter (t), then (f(\gamma(t))) must be a geodesic with parameter (t) (not just a reparameterized version of a geodesic). This ensures the map respects the "straightest path" structure defined by each connection.

Local Coordinate Calculation #

If you're working with explicit coordinates, you can translate the core condition into a formula involving Christoffel symbols:

• Let ((x^i)) be local coordinates on (M) with Christoffel symbols (\Gamma^k_{ij}) for (D), and ((y^\alpha)) be local coordinates on (M') with Christoffel symbols (\Gamma'^\gamma_{\alpha\beta}) for (D'). For (f) to preserve the connections, the following must hold for all indices (i,j,k,\alpha,\beta,\gamma):
  [
  \frac{\partial f^\gamma}{\partial x^k} \Gamma^k_{ij} = \Gamma'^\gamma_{\alpha\beta} \frac{\partial f^\alpha}{\partial x^i} \frac{\partial f^\beta}{\partial x^j} + \frac{\partial^2 f^\gamma}{\partial x^i \partial x^j}
  ]
  This is just the coordinate-based expansion of the pushforward-covariant derivative commutation rule, useful for hands-on computations.

All three of these conditions are equivalent, so you can pick the one that makes the most sense for your current work—whether that's abstract theory, geometric reasoning, or concrete calculations.

内容的提问来源于stack exchange，提问作者Hussein Eid