Here is another possible explanation.
Theorem of Govorov-Lazard: Let $A$ be a ring. Then an $A$-module is flat if and only if $M$ is a filtered colimit of finite free $A$-modules.
See MO/127769 for applications of this Theorem. A Riemannian manifold is called flat if its curvature is $0$ i.e. locally it looks like affine $n$-space $\mathbb{R}^n$. If $A$ is commutative, the functor $M \mapsto \mathbb{V}(M) := \mathrm{Spec}(\mathrm{Sym}(M))$ from $A$-modules to $A$-schemes maps finite free $A$-modules onto affine spaces over $A$. Hence, it maps flat $A$-modules onto filtered limits of affine spaces (where the transition maps should be "linear"). So we definitely get a (vague) connection between flat modules and flat manifolds. It has been discussed at MO/19308 if there is a notion of curvature in algebraic geometry.
