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

## TL;DR

This paper introduces a distributional approach to derive Pizzetti and Cauchy formulas for higher-dimensional surfaces, providing new proofs and interpretations for integration on manifolds and operators like the Dirac operator.

## Contribution

It develops a general distributional method for manifold integration and applies it to derive Pizzetti formulas and a Cauchy theorem for the tangential Dirac operator.

## Key findings

- Derived an alternative proof of Pizzetti formulas for Stiefel manifolds
- Provided a distributional interpretation of invariant oriented integration
- Established a distributional Cauchy theorem for the tangential Dirac operator

## Abstract

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.