An application of the h-principle to manifold calculus

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor Emb _Lag(- , N) is the totally real embeddings functor Emb _TR(- , N). More generally, for subsets A of the m-plane Grassmannian bundle Gr(m,TN) for which the h-principle holds for A-directed embeddings, we prove the analyticity of the A-directed embeddings functor EmbA(-,N).