Real analytic extension of functions on normal crossings

TL;DR

This paper proves that functions which are real analytic on each component of a normal crossings union in a compact manifold can be extended to a real analytic function on the entire manifold, using tools from real analytic geometry.

Contribution

It establishes the extension of real analytic functions from unions with normal crossings to the whole manifold, employing Cartan's theorems A and B.

Findings

Extension of real analytic functions across normal crossings unions

Use of Cartan Theorems A and B in the proof

Applicable to compact real analytic manifolds

Abstract

We consider a compact $C^\omega$ manifold $X$ and finitely many regular $C^\omega$ submanifolds $Y_1, \dots, Y_q$ of $X$, which are closed subsets in $X$, such that the union of $Y_j$'s has only normal crossings. We show that every continuous function on the union which is of class $C^\omega$ on each $Y_j$ can be extended to a $C^\omega$ function on $X$. A crucial feature of our proof is to employ basic tools of real analytic geometry -- Cartan Theorems A and B.