Global fiber and local differential criteria for Bolker

A tangent vector to $Z$ at $(g,x(g,s))$ has the form $(\dot{g},D_{g}x\dot{g}+D_{s}x\dot{s})\mathrm{.}$ A pair $(\eta ,-\xi )$ annihilates every such tangent vector exactly when $(D_{s}x{)}^T\xi =0$ and $\eta =(D_{g}x{)}^T\xi$. This follows by setting $\dot{g}=0$ and then varying $\dot{g}$ arbitrarily. Thus all conormal pairs, except the zero pair, are parameterized by $(g,s,\xi )\in E_g$. The rank condition on $D_{s}x$ makes this conormal parameterization smooth.

Under this parameterization, the left projection sends $(g,s,\xi )$ to $(g,(D_{g}x{)}^T\xi )$, as stated. Because the first component is retained, two points in this chart portion have the same left image exactly when their $g$ coordinates agree and their values under $L_g$ agree. This proves the chart-portion injectivity criterion; it says nothing about a second preimage outside this chart.

A tangent vector in the kernel of the differential of the left projection has $\dot{g}=0$. Differentiating the defining equation $(D_{s}x{)}^T\xi =0$ along $(\dot{s},\dot{\xi })$ gives $(D_{s}x{)}^T\dot{\xi }+(D_s^{2}x[\dot{s},\cdot ]{)}^T\xi =0.$ Differentiating its output $(D_{g}x{)}^T\xi$ gives the second displayed equation in the statement. Therefore the differential is injective exactly when the simultaneous system has only the zero solution. This is the asserted rank test, and by [F2] it is precisely the local immersion clause of Bolker.