Differential topology

I thinks there is some problem related to the solution of a 2 order differential equation given by Sturm-Liouville system which is nontrivial. It is well-know that the power of Sturm-Liouville theory see  wiki, is due to it is some kind of “spectral decomposition” in the solution space. Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURM-LIOUVILLE. PROBLEMS BY APPROXIMAT…

https://en.wikipedia.org/wiki/Transversality_(mathematics) Question 1: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+1}$, and take a point $p \notin M$. Prove that there is always a line $l_p$ passing through $p$ such that $l_p \cap M \neq \emptyset$, and $l_p$ intersects transversally with $M$. Question 2: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+m}$, and take a point $p \notin M$. Prove that for all $1\leq k\leq m$, there is always a hyper…