On uniqueness of the equivariant smooth structure on a real moment-angle manifold

The paper is devoted to the well-known problem of smooth structures on moment-angle manifolds. Each real or complex moment-angle manifold has an equivariant smooth structure given by an intersection of quadrics corresponding to a geometric realisation of a polytope. In 2006 F.Bosio and L.Meersseman proved that complex moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic. Using arguments from calculus we derive from this result that real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic and the polytopes are diffeomorphic as manifolds with corners.