# 3 manifolds – Parametric general position theorem for foliations

The situation is the following: let $$M$$ be a manifold endowed with a smooth foliation $$mathcal{F}$$ of codimension one (suppose orientable, transversely orientable) and let $$F_t : S rightarrow M$$ be a one parameter family of smooth inmersions. Is it true that there exists an arbitrarily small homotopy from $$F_t$$ to a family of inmersions $$G_t$$ such that $$G_t$$ is in general position with respect to $$mathcal{F}$$? Furthermore can I say something about the number of singularities at each step and how are the transitions? (maybe this is a lot to ask). More concretely I’m thinking in $$M = mathbb{T}^3$$ and the embedded surfaces being $$mathbb{T}^2 times {pt}$$.

Pd: by general position I mean that it is transversal except at finitely many points where the induced foliation on $$S$$ has Morse type singularities.