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 \(\mathrm {Emb}_{\mathrm {Lag}}(-,N)\) is the totally real embeddings functor \(\mathrm {Emb}_{\mathrm {TR}}(-,N)\). More generally, for subsets \({\mathcal {A}}\) of the m-plane Grassmannian bundle \({{\,\mathrm{{Gr}}\,}}(m,TN)\) for which the h-principle holds for \({\mathcal {A}}\)-directed embeddings, we prove the analyticity of the \({\mathcal {A}}\)-directed embeddings functor \({{\,\mathrm{Emb}\,}}_{{\mathcal {A}}}(-,N)\).