Let $M$ be a smooth manifold and let $Q$ be an arbitrary $C^\infty(M)$-module. $Q$ is called geometric if $$\bigcap_{p\in M}\mu_pQ=0,$$ where $\mu_p$ is an ideal in $C^\infty(M)$ of functions vanishing at point $p.$

From the definition, $q\in\mu_pQ$ iff $q=\sum_{i=1}^Nf_iq_i$ for some $f_i$'s from $\mu_p$ and $q_i$'s from $Q.$

Request.Construct a non-geometric $C^\infty(M)$-module.

For any vector bundle $E\to M$ the $C^\infty(M)$-module $\Gamma(E)$ is geometric. This is because, if $\xi\in\mu_p\Gamma(E),$ then $\xi(p)=0.$ Hence we have to look for non-geometric modules in a different way.

I came across this notion in the following Jet Nestruev's book:

*Nestruev, Jet*, **Smooth manifolds and observables**, Graduate Texts in Mathematics. 220. New York, NY: Springer. xiv, 222 p. (2003). ZBL1021.58001.

**Remark about the notion:**
I investigated more or less Vinogradov's bibliography and I guess this notion first appeared in the following book:

*Krasil’shchik, I.S.; Lychagin, V.V.; Vinogradov, A.M.*, Geometry of jet spaces and nonlinear partial differential equations. Transl. from the Russian by A. B. Sosinskij, Advanced Studies in Contemporary Mathematics, 1. New York etc.: Gordon and Breach Science Publishers. xx, 441 p. (1986). ZBL0722.35001.