Suppose \(M\) is a smooth manifold, \(\nabla\) is a connection on and \(\gamma\) denotes a smooth curve on \(M\) starting in \(p\in M\). Then, by some ODE theory, it follows that for a given \(v\in T_pM\) there exist a unique vector field \(V\) along \(\gamma\) such that

\[ V_p = v \]

[1, Theorem 4.32].

Therefore, we can define the parallel transport map by

\[ P_{t_0t_1}^{\gamma}\colon T_{\gamma(t_0)}M\to T_{\gamma(t_1)}M \]

with \(P_{t_0t_1}^\gamma(v)=V(t_1)\).

The map is a linear isomorphism.

Links Link to heading

• parallel vector field
• connections are derivatives
• interpretation of curvature via parallel transport

References Link to heading

1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9