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?


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