Atiyah and Todd classes arising from integrable distributions

In this paper, we study the Atiyah class and Todd class of the DG manifold $(F[1],d_F)$ corresponding to an integrable distribution $F \subset T_{$\mathbb{K}$} M = TM \otimes_{$\mathbb{R}$} $\mathbb{K}$$,where$$\mathbb{K}$ = $\mathbb{R}$$or$$\mathbb{C}$$. We show that these two classes are canonically identical to those of the Lie pair $(T_{\mathbb{K}}M,F)$. As a consequence, the Atiyah class of a complex manifold $X$ is isomorphic to the Atiyah class of the corresponding DG manifold $(T_X^{0,1}[1],\bar{\mathrm{\partial }})$. Moreover, if $X$ is a compact K"ahler manifold, then the Todd class of $X$ is also isomorphic to the Todd class of the corresponding DG manifold $(T_X^{0,1}[1],\bar{\mathrm{\partial }})$.