I am dealing with the following exercise:
Let $\sigma=(\sigma_1, \sigma_2) : (0, 1) \to R^2$ be a smooth curve such that $ ||\sigma ' (t)|| =1 $ and $\sigma_1(t) >0$ for any $t \in (0, 1)$. Consider $G: (0,1) \times R^2 \to R^3$ given by $$G(t, x_1, x_2)=(\sigma_1(t)x_1, \sigma_1(t)x_2, \sigma_2(t)).$$
Finally, consider $F$ the restriction of $G$ to $(0, 1) \times S^1$.
I have to prove that $F$ is an immersion.
My idea was to consider the Jacobian determinant of $G$, and to see if it is different from $0$ on $(0, 1) \times S^1$. This would fulfil the request, right? However, I find that $$ \text{det} JF= \sigma_2 '(t) \sigma_1(t)^2,$$ and I can't see how this quantity should be different from $0$. Or is there a different way to prove $F$ to be an immersion?
I thank you in advance for any kind of help.
If you were asked to prove that $G : (0,1) \times R^2 \to R^3$ was an immersion, then computing the $3 \times 3$ Jacobian determinant would be the way to go. But, that is not what you were asked to prove.
Consider the derivative map $dG$ at a point $p = (t,x_1,x_2)$, a linear map from the tangent space of $(0,1) \times R^2$ at $p$ to the tangent space of $R^3$ at $G(p)$. What are are asked to do is to prove that if $p \in (0,1) \times S^1$ (i.e. if $x_1^2+x_2^2=1$) then $dG$ restricts to an injective map from the 2-dimensional tangent space of $(0,1) \times S^1$ at $p$ to the 3-dimensional tangent space of $R^3$ at $G(p)$. One thing you can do is to work out a basis for that 2-dimensional tangent plane, and then work out the $3 \times 2$ matrix for the restriction of $dG$ to that tangent plane, and then prove that this matrix has rank 2.