Hello I was trying to answer the question:
Let $S$ be a compact, connected, smooth regular surface and $a$ be a unit vector in $\mathbb{R}^3.$ Prove that there exists a point on $S$ whose normal line is parallel to $a.$
I was trying to prove it in the following way:
Suppose there is a an $a$ such that no normal line is parallel to $a.$ This implies that for all tangent planes $T_p,$ $a$ is not normal to $T_p.$ So, there is no $T_p$ of the form
$$P_{x_0}=\{x\in \mathbb{R}^3:\langle x-x_0,a\rangle=0\}.$$
Hence, if $P_{x_0}$ intersects $S,$ then it must intersect at least $2$ points of $S.$ Let $P_{x_0}$ be the plane which intersects $S$ at the minimum numbers of points (but not zero).
Now here, I was wondering if I could bring in the property of the compactness of the surface, but I got stuck. I also have absolutely no idea how to bring in connectedness into the picture here as well.
How should I go about this proof? Any help will be much appreciated, it would mean a lot!
HINT: Because the surface is compact, every continuous real-valued function on it has a maximum (and minimum). Consider the smooth function $f\colon S\to\Bbb R$ defined by $f(x)=x\cdot a$.
Comment: Connectedness is irrelevant. Can you see why?