# Checking real analyticity on surfaces

## TL;DR

The paper proves that a real-valued function on a real analytic surface is analytic if its restrictions to all 2-sphere submanifolds are analytic, extending a classical complex analysis result to real manifolds.

## Contribution

It establishes a new criterion for analyticity of functions on real surfaces based on restrictions to 2-spheres, analogous to Hartogs' theorem in complex analysis.

## Key findings

- Functions are analytic if restrictions to all 2-sphere submanifolds are analytic.
- No continuity assumption is needed for the function.
- Extends classical Hartogs theorem to real analytic manifolds.

## Abstract

We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the 2-sphere are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is complex analytic iff it is complex analytic when restricted to any complex curve.