Some nonlocal formulas inspired by an identity of James Simon

TL;DR

This paper introduces a new nonlocal geometric formula inspired by James Simons' classical identity, which generalizes to fractional settings and recovers the classical case, offering a differential geometry-independent approach.

Contribution

The paper presents a novel nonlocal geometric formula inspired by Simons' identity, extending it to fractional contexts and providing a new perspective beyond traditional differential geometry.

Findings

Derived a new nonlocal geometric formula

Generalized classical identity to fractional settings

Recovers classical case in the limit

Abstract

Inspired by a classical identity proved by James Simons, we establish a new geometric formula in a nonlocal, possibly fractional, setting. Our formula also recovers the classical case in the limit, thus providing an approach to Simons' work that does not heavily rely on differential geometry.