Title: An Application of $h$-principle to Manifold Calculus,
Abstract: 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 we apply the theory of
h-principle to construct several examples of analytic functors in this sense.
We prove that the analytic approximation of the Lagrangian embeddings functor
$\mathrm{emb}_{\mathrm{Lag}}(-,N)$ is the totally real embeddings functor
$\mathrm{emb}_{\mathrm{TR}}(-,N)$. Under certain conditions we provide a
geometric construction for the homotopy fiber of $ \mathrm{emb}(M,N)
\rightarrow \mathrm{imm}(M,N)$. This construction also provides an example of a
functor which is itself empty when evaluated on most manifolds but it's
analytic approximation is almost always non-empty.