Let $M$ be a smooth manifold and $x:M\rightarrow \mathbb{R}$ a smooth function with $0$ as regular value, such that $X=\{x=0\}\subset M$ is a smooth submanifold. Then $$ 0\rightarrow x C^\infty(M)\hookrightarrow C^\infty(M) \xrightarrow{f\mapsto f\vert_X} C^\infty(X)\rightarrow 0 $$ is a short exact sequence and a right split corresponds to an extension map.

Where does this show up:

• For $M=\mathbb{R}$ this is the statement that the remainder in a Taylor series can be written as $R(x)=x^kr(x)$ for a smooth function $r(x)$.
• For $M=\mathbb{R}^d\times \bar {\mathbb{R}}^d$ (where $\bar{\mathbb{R}}^d$ is the radial compactification) and $x$ a boundary defining function of $\partial M=\mathbb{R}^d\times S^{d-1} = S^*\mathbb{R}^d$ (co-sphere bundle), this yields $$ 0\rightarrow \Psi_{\mathrm{cl}}^{m-1}(\mathbb{R}^d)\hookrightarrow \Psi_{\mathrm{cl}}^{m}(\mathbb{R}^d) \xrightarrow{\sigma_m} C^\infty(S^*\mathbb{R}^d)\rightarrow 0, $$ the shorth exact symbol sequence of pseudo-differential operators. Here $\Psi^m(\mathbb{R}^d)=\mathrm{Op}(x^{-m}C^\infty(M))$ with $\mathrm{Op}$ denoting the standard quantisation of symbols $a:\mathbb{R}^d_z\times \mathbb{R}^d_\xi\rightarrow \mathbb{C}$. One can take $x=\langle \xi \rangle^{-1}$ as bdf. of fibre-infinity. A right split is then a quantisation map. The symbol sequence (together with the multiplicativity of $\sigma_m$) allows to construct parametrices of elliptic operators and is thus the starting point of elliptic regularity theory.
• For $M=\bar {\mathbb{R}}^d\times \bar {\mathbb{R}}^d$, which is a manifold with corners, the constructions from the previous point yields Melrose's scattering (classical) scattering pseudo-differential operators.