I’m reading the book “Differential Geometry: Cartan’s Generalisation of Klein’s Erlangen Program” from Sharpe. Given a reductive model geometry, where $ \mathfrak g=\mathfrak h\oplus \mathfrak p$ with $ \mathfrak p=\mathfrak{g/h}$ (where $ \mathfrak p$ need not be Lie subalgebra of $ \mathfrak g$ , i.e closed under the Lie bracket), every $ \mathfrak g$ -valued form decomposes into $ \mathfrak h$ and $ \mathfrak p$ components. Moreover, this also holds for the universal covariant derivative, i.e $ \tilde D_X=\tilde D_{\mathfrak h}{}_X+\tilde D_{\mathfrak p}{}_X$ . From the Cartan connection decomposition, we get the Ehresmann connection $ \omega_\mathfrak{h}:TP\to\mathfrak h$ , which lies in the total space of the Cartan geometry $ (P,\omega)$ , i.e the principal (right) fibre bundle with total space $ P$ , base space $ M$ and structure group $ H$ . I have studied such connections in the past and also the method one uses to construct the usual covariant derivative from these. The way I have seen includes amongst other steps the existence of a bijective correspondence between sections of the associated (to $ P$ ) bundle with total space $ E=(P\times F)/H\equiv P\times_H F$ , i.e $ \sigma:M\to F$ and $ H$ -equivariant functions $ \phi:P\to F$ . Choosing a local section of $ P$ , $ \varphi:U\to P$ and a local section of $ E$ , $ s:U\to F$ , one finally comes to the result $ \nabla_Xs=(\mathrm ds+\varphi^*\omega_{\mathfrak h})(X)$ , where $ \nabla_Xs=(\varphi^*D\phi)(X)$ with $ D\phi$ being the covariant exterior derivative of the 0-form $ \phi$ , given by $ D\phi(T)=\mathrm d\phi(T)+\omega_{\mathfrak h}(T)\phi$ for a vector field $ T$ on $ P$ .

Now, Sharpe concludes that $ \tilde D_\mathfrak{p}$ is the usual covariant derivative with $ D_X\phi=X(\phi)-\rho_*(\theta_\mathfrak{h}(X))\psi$ , where $ \psi:\Gamma(E)\to \Omega^0P\otimes(V,\rho)$ , $ \rho_*:\mathfrak h\to \mathrm{End}(V)$ and $ \phi$ is the expression of a tensor $ \Phi$ of type $ (V,\rho)$ in the cartan Gauge $ (U,\theta)$ , i.e $ \phi=\Phi\circ\sigma:U\to V$ with $ \sigma$ being a local section of $ P$ and $ R_h^*\phi=\rho(h^{-1})\phi$ . I’m now trying to compare. It seems to me that Sharpe’s $ D_X\phi$ must be the same as $ \nabla_Xs$ , since both $ \phi$ and $ \sigma$ are local sections of the associated vector bundle (Sharpe considers the total space $ E=P\times_H (V,\rho)$ ). We need $ D_X\phi$ or $ \nabla_Xs$ to be again a section of the associated bundle. In my first definition, this is the case indeed. In Sharpe’s definition I cannot see how this holds. Firstly, there is the problem of the fibre of the associated bundle. Is it $ V$ or $ (V,\rho)$ ? If it is $ V$ , then $ \psi$ must be $ \phi$ (typo of the author then?) and the claim is valid. Otherwise, I can’t make any sense. Furthermore, how do both definitions relate? Shouldn’t they be equivalent?