Manifolds are often defined as embedded submanifolds of Euclidean space Rn, so this forms a very important special case.
He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in , and a new proof of Remmert's proper mapping theorem.
The methods apply equally well to domains such as soap films, fractals, charged particles, and Whitney stratified spaces, placing them on the same footing as smooth submanifolds in the resulting calculus.
Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.