# The Transversality Condition is Generic

Let $$M$$ and $$N$$ be submanifolds of $$\mathbb{R}^n$$. While I understand that the transversality condition $$M \pitchfork N$$ is stable (assuming $$M$$ is compact), I want to show the following property (without assuming $$M$$ is compact):

If there exists a point $$p\in M\cap N$$ such that $$T_pM+T_pN=\mathbb{R}^n$$, there exists a neighborhood $$U$$ (possibly in $$M\cap N$$) containing $$p$$ such that $$T_qM+T_qN=\mathbb{R}^n$$ for all $$q\in U$$.

While I intuitively see/believe this is true, is there a way to formalize a proof using the stability of transversality? I first attempted to consider linear independence in $$T_pM$$, but I'm not sure how to continuously relate this linear independence to "nearby" tangent spaces. Thank you for any help.

• Reduce this immediately to a regular value statement and use the continuity of determinant to infer that submersions are stable. Commented Aug 2, 2023 at 17:07
• Thank you for your comment — I fleshed out your answer in a response! Commented Aug 3, 2023 at 16:55
• Well done. I wish all questioners were as pro-active with my hints :) Commented Aug 3, 2023 at 18:10

Since $$N$$ is embedded, we can use slice coordinates to (locally) write $$N$$ as the zero set of independent functions $$x^{k+1},\dots ,x^{n}$$. We have $$\iota^{-1}(N)=M\cap N$$, so it follows that $$M\cap N$$ is locally the vanishing set of $$x^{k+1}\circ \iota,\dots,x^n\circ \iota$$.
Write $$g=(x^{k+1},\dots ,x^{n})$$. To apply the preimage theorem, it can be shown that zero is a regular value of $$g\circ \iota$$ if and only if
$$T_q M+T_q N=T_q\mathbb{R}^n\cong\mathbb{R}^n$$
for all $$q\in M\cap N$$. Suppose that this equation holds at some $$p\in M\cap N$$, so that $$d(g\circ \iota)_p$$ is surjective. It is clear this matter is local in both the domain and codomain, so we reduce to the case where $$M$$ is an open set in $$\mathbb{R}^k$$, and the codomain is an open set in $$\mathbb{R}^m$$. Here, surjectivity gives $$k\geq m$$, and equivalently, that $$d(g\circ \iota)_p$$ must have an invertible $$m\times m$$ submatrix. Call the column indices of this submatrix $$i_1,\dots,i_m$$. For example, if $$d(g\circ \iota)_p=\begin{pmatrix}1 & 1 & 0\\ 0 & 1 & 1\end{pmatrix}$$, then we are selecting $$i_1,i_3$$.
Taking the determinant of this submatrix, continuity yields a neighborhood $$U$$ about $$p\in M$$ for which the determinant of the submatrix with column indices $$i_1,\dots,i_m$$ from the matrix $$d(g\circ \iota)_q$$ is non-zero. Thus, $$d(g\circ \iota)_q$$ is surjective for all $$q\in U$$, and so the equation $$T_q M+T_q N=\mathbb{R}^n$$ holds for all such points.