Might be a strange question, but what textbook/reference should I read in order to be able to solve problems like the followings? I only took one class in classical differential geometry, and we covered chapter one through chapter five from this notes. But still I do not have sufficient knowledge to solve these problems.
I have taken standard (?) undergraduate level classes in analysis (Apostol, but no measure theory), topology (Greene) and algebra (Hungerford, the introduction one).
Example 1: Let $S^2$ be the unit sphere in $\mathbb{R}^3$. Define map $f: S^2 \to \mathbb{R}^3$ by $$f (x,y,z)\longrightarrow (yz-x,zx-y,xy-z)$$ Determine all the singular points of $f$. A point $p$ is singular if if the rank of the differential at $p$ is less than 2.
Example 2: Let $n \ge 1$ be an integer and $M \subset \mathbb{R}^{n+2}$ a smooth $n$-dimensional submanifold, which is a closed subset of $\mathbb{R}^{n+2}$. Prove that for any $x_0 \in M$, there exists a line $L$ in $\mathbb{R}^{n+2}$ satisfying the condition: $$L\cap M = \{x_0\}$$
Example 3: Let $V$ be an $n$-dimensional real vector space with $n \ge 2$. Prove that every element $v \in \wedge^{n−1}V$ can be written as $$v = v_1 \wedge ... \wedge v_{n−1}$$ with some elements $v_1, ..., v_{n−1} \in V$ . Here $\wedge^{n−1}V$ denotes the $(n − 1)$-st exterior product of $V$ .