A question on the completely positive maps and manifold structure

Question

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a completely positive map such that $\sum_{j=1}^nV_jV_j^*\leq I$ (i.e. a contraction). Then the curvature is defined as \begin{equation} K(\Phi):=(n-1)\lim_{d\rightarrow\infty}\frac{Tr(I-\Phi^d(I))}{n^d} \end{equation} and the Euler characteristic is defined as \begin{equation} \chi(\Phi):=(n-1)\lim_{d\rightarrow\infty}\frac{Rank(I-\Phi^d(I))}{n^d}. \end{equation} My problem is, I can not see how the above quantities define the underlying geometry. Further, I am not sure, whether the set of completely positive maps (in finite or infinite dimensions) can form a manifold. I am familiar with Euler characteristic defined in topology (and graph theory), but can not get the proper intuition to connect it in this scenario. Please help.