Let $S^2 \subset E^3$ be the unit $2$-sphere in Euclidean $3$-space. Set $M = \{p_1, p_2 \in S^2 : p_1 \neq p_2\}$.
Define $f : M \to \mathbb{R}$ by setting $f(p_1, p_2)$ to be the square of the Euclidean distance between $p_1, p_2 \in S^2 \subset E^3$. Prove that $f$ is smooth.
$f(p_1,p_2) = (x_{p_1} - x_{p_2})^2 + (y_{p_1} - y_{p_2})^2 + (z_{p_1} - z_{p_2})^2$, which is a composition of basic smooth functions, hence is smooth.
Is this good then?
And furthermore,
How can I show that $S = f^{-1}(1/2) \subset M$ is a submanifold? What is the dimension of $S$? Is $S$ compact? Bonus: Can you identify $S$?
Certainly I need to show that $1/2$ is a regular value. And that is to show $$2(x_{p_1} - x_{p_2}) + 2(y_{p_1} - y_{p_2}) + (z_{p_1} - z_{p_2}) \neq 0.$$ But I found a hard time to show this algebraicly.
I tried to parametrize $p_1, p_2$ such that $p_1 = \gamma(0)$ and $p_2 = \gamma(t)$, but then I am not sure about differentiating $f(\gamma(t_1, t_2))$: $$\frac{df}{dt} = df \circ d \gamma.$$
I hope to show that $df \neq 0$ at $\gamma(t)$ and $d \gamma(t) \neq 0$ at $t$. Not sure if this is the right approach, nor how to do it.
As well as following questions on the dimension, compact, and identify.. Some hints would be really great.
Thank you very much!