Pizzetti and Cauchy formulae for higher dimensional surfaces: a distributional approach

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 $φ_1(\underline{x})=\ldots=φ_k(\underline{x})=0$. 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.

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