Background for 2 differential geometry questions I encountered a couple of questions in a collection of differential geometry exams that I don't know how to approach. Of course I am NOT expecting a solution to these, but just a hint.


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*If $S\subset \mathbb{R}^3$ is a non-empty surface, show that there exists a plane $ax+by+cz=d$ in $\mathbb{R}^3$ whose intersection with $S$ is a collection of circles and lines.

*If $M$ is an n-dimensional manifold embedded in $\mathbb{R}^n$ and $k\geq m-n$, then there exists a k-hyperplane $P$ such that $P\cap M$ is a non-empty smooth manifold of dimension $n+k-m$.
As I mentioned, I would simply like to know what is being tested.
 A: It sounds like these questions are testing knowledge of some kind of pre-image theorems. Here are a couple to give you an idea of what sort of thing might be being tested:
$1)$If $X$ and $Y$ are smooth manifolds without boundary, $f:X\rightarrow Y$ a smooth map between them and $y$ a regular value of $y$, then $f^{-1}(y)$ is a smooth submanifold of $X$ of dimension $\dim(X)-\dim(Y)$.
$2)$If $X$ is a smooth manifold with boundary, $Y$ a smooth manifold without boundary and $f:X\rightarrow Y$ is a smooth map, if $y \in Y$ is a regular value for $f$ and $f|_{\partial X}$ then $f^{-1}(y)$ is a smooth $\dim(X)-\dim(Y)$ manifold with boundary $\partial X\cap f^{-1}(y)$.
There are a few more, all in this vein. I think $1$ should be enough to do both questions (combined with the classification of $1$-manifolds).
Of course, it may also be testing knowledge of transversality. A simple case of this is as follows: We say that $X$ and $Y$, two submanifolds of a manifold $Z$, intersect transversally if for any $x \in X\cap Y$, we have $T_xX+T_xY=T_xZ$. We have:
$3)$ If $X$ intersects $Y$ transversally, then $X\cap Y$ is a submanifold of $Z$ with $\operatorname{codim}(X\cap Y)=\operatorname{codim}(X)+\operatorname{codim}(Y)$.
I'm fairly sure you could do both problems with either $1)$ or $3)$, with $1)$ you would take the function $(x,y,z)\mapsto ax+by+cz$ for various $a,b,c$ and find a regular value $d$, with $3)$ you would find a plane with transverse intersection. These two ideas are highly related when it comes to planes, since planes are just preimages of regular functions between $\mathbb{R}^n$ and $\mathbb{R}$, so intersecting with a plane is the same as taking the preimage of the regular function that determines the plane. If I had to guess I would assume the course is testing pre-image theorems, since I would say that they are more "basic" in some sense than transversality.
