  In this paper, we study Pizzetti-type formulas for Stiefel manifolds and
Cauchy-type formulas for the tangential Dirac operator from a distributional
perspective. First we illustrate a general distributional method for
integration over manifolds in $\mathbb R^m$ defined by means of $k$ equations
$\varphi_1(\underline{x})=\ldots=\varphi_k(\underline{x})=0$. Next, we apply
this method to derive an alternative proof of the Pizzetti formulae for the
real Stiefel manifolds $SO(m)/SO(m-k)$. Besides, a distributional
interpretation to invariant oriented integration is provided. In particular, we
obtain a distributional Cauchy theorem for the tangential Dirac operator on an
embedded $(m-k)$-dimensional smooth surface.
