Real Analyticity is Concentrated in Dimension 2

Abstract

We prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.