On Milnor's Morse theory, we have:

Given a manifold $M\subset\mathbb{R}^n$, let $N=\{(q,v)~\vert~ q\in M,$ $v$ is perpendicular to $M$ at $q\}\subset M\times \mathbb{R}^n$

And $E:(q,v)\mapsto q+v$.

A point $e\in \mathbb{R}^n$ is a focal point of a manifold $M\subset \mathbb{R}^n$ if $e=q+v$ for some $(q,v)\in N$ and the Jacobian of $E$ at $(q,v)$ has nullity $>0$.

My question is, Milnor claims that a focal point of M is a point in $\mathbb{R}^n$ where nearby normals intersect, how to see this claim?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.