# Genericity of maps which are transverse when restricted to a submanifold

Let $M$ and $N$ be smooth manifolds, and $A\subset M$, $B\subset N$ be smooth embedded submanifolds. I am looking for a reference for a theorem on the following lines:

The set of smooth maps $h\in C^\infty(M,N)$ such that $h\bigr|_{A}\pitchfork B$ is generic, i.e. it contains a countable intersection of open and dense subsets, in both the strong and weak topologies.

With the transversality assumption I really mean that the map $h\bigr|_{A}:A\rightarrow N$ is transverse to $B$. Thus I want that for each $a\in A$, such that $h(a)\in B$ that $dh T_aA+T_{h(a)}B=T_{h(a)}N$.

I expect that this is true. I have a sketch of a proof of a slightly more simple situation, which is where I need to apply the theorem, but the details are a bit messy. My question is therefore: Where can I find this (or a similar) theorem in the literature?

• Hirsch's Differential Topology, chapter II on transversality, Theorem 2.1(b) Apr 11, 2014 at 19:17
• @Chris Gerig, thanks for you comment. I know of this theorem, but this is not what I want. I don't want that for every $a\in A$, $h\pitchfork B$ at $a$, but I want that the restriction of $h$ to the submanifold $A$ is transverse. These are two different things. Apr 11, 2014 at 19:43

Given Chris Greig's reference, you just need to show that for the restriction map $C^\infty(M,N)\to C^\infty(A,N)$ the inverse image of an open dense set is open and dense. I presume this is written down somewhere, maybe even in Hirsch, but in any case is easy to show.