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 S2 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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Homogeneous polynomial
- Real analytic function
- Real analytic manifold