This is a follow-up (refinement?) of this question.
In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of terminology because it provides an embedding of $Rings^{op}$ into a larger category.) One construction we can do, for example, is create the "tangent scheme", which is obtained by localizing and completing. The examples I've been looking at are $\hat{\mathbb{G}}_a$, $\hat{\mathbb{G}}_m$, $(\mathbb{Z}[[t]],F)$ (FGLs (1-dimensional, commutative) more generally), $T_1C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_m)\cong C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_a)$ ("commutative $k$-variate FGLs satisfying the 2-cocycle condition"), etc. We then have exponential maps, which e.g. in the last case is $exp:C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_a) \rightarrow C^k(\hat{\mathbb{G}}_a,\hat{\mathbb{G}}_m)$ given by $g\mapsto 1+g$.
Just to test the waters, here is my question (although please feel free to push it further or in a different direction). Presumably these exponential maps are not always injective. Once we apply this picture to a particular ring (or perhaps even before?), can we translate differential-geometric ideas like conjugate points, geodesics, "$exp$ is a local isomorphism", etc. into algebro-geometric language? When we can, which theorems for manifolds carry over to schemes and which must we discard?