I'm working on two exercises from Guillemin-Pollack which have the same flavor:
(General Position Lemma) Let $X$ and $Y$ be submanifolds of $\mathbb R^N$. Show that for almost
every $a\in \mathbb R^N$, the translate $X+a$ intersects $Y$
transversally.
Suppose that $X$ is a submanifold of $\mathbb R^N$. Show that almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally. [HINT: The set $S\subset (\mathbb R^N)^l$ consisting of all linearly independent $l$-tuples if vectors in $\mathbb R^N$ is open in $\mathbb R^{Nl}$, and the map $\mathbb R^l\times S\rightarrow \mathbb R^N$ defined by $[(t_1,\dots,t_l),v_1,\dots,v_l]\mapsto t_1v_1+\dots t_lv_l$ is a submersion.]
In both cases, I believe I need to apply the following version of the Transversality Theorem (see this answer):
Theorem: Suppose that $F:X\times S\to Y$ is a smooth map of manifolds and $Z$ is a submanifold of $Y$, all manifolds without boundary. If $F$ is transverse to $Z$ then for almost every $s\in S$ the map $f_s : x\mapsto F(x,s)$ is transverse to $Z$.
I have the same problem in both exercises.
- In the first exercise, the theorem guarantees that for almost every $a\in \mathbb R^N$, the map $f_a: X\rightarrow \mathbb R^N$ given by $x\mapsto x+a$ is transversal to $Y$. Note that the image of this map is $X+a$. I need to show that the image is transversal to $Y$. How does it follow?
- In the second exercise, the theorem guarantees that for almost every $v=(v_1,\dots,v_l)\in S$, the map $f_v: \mathbb R^l\rightarrow \mathbb R^N$ given by $(t_1,\dots,t_l)\mapsto t_1v_1+\dots+t_lv_l$ is transversal to $X$. Note that the image of this map is an $l$-dimensional subspace of $\mathbb R^N$. I need to show that the image is transversal to $X$. How does it follow?
Best Answer
Let $i_1 : M_1 \to X$ and $i_2 : M_2 \to X$ be embeddings. The following statements are equivalent:
This follows from the fact that for an embedding $f : M \to X$, $f_*(T_pM) = T_{f(p)}(f(M))$.