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 $\mathrm{Emb}_{\mathcal{A}}(-,N)$.