Let \(M\) be a smooth manifold and \(\nabla\) a connection on \(M\). Then for \(X,Y\in \mathfrak{X}(M)\) \(p\in M\)

\[ \nabla_X Y|_p = \lim_{h \to 0} \frac{P_{h0}^\gamma Y_{\gamma(h)}-Y_p}{h}, \]

where \(\gamma\) is any smooth curve on \(M\) satisfying \(\gamma(0)=p\) and \(\gamma'(0)=X_p\) [1, Theorem 4.35].

Links Link to heading

• connection
• 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