In this lecture we give two additional ways to view the curvature of a connection $\nabla$ on a vector bundle $\pi \colon E \to M$.

The first is geometric in nature, and roughly speaking says that curvature can be computed by taking a time derivative of parallel transport around shorter and shorter loops. Or in symbols:

$$R^{\nabla}(\xi,\zeta)(v) = -\mathcal{J}_v^{-1} \! \left( \frac{d}{dt}\Big|_{t=0}\mathbb{P}_{\eta_t}(v) \right).$$

This formula is not particularly useful as far as computations are concerned. The second part of the lecture is devoted to established a rather more useful formula — albeit less geometric — which reads:

$$R^\nabla(X,Y)(s) = \nabla_X\nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X,Y]}s.$$

We also use the opportunity to state the famous Ambrose-Singer Holonomy Theorem, whose proof will come later in the course.