Question: Exercise 1, Section 4.2, Do Carmo Differential Geometry of Curves and Surfaces:
Let F be a parametrization $F: U\in \mathbb{R}^2\rightarrow \mathbb{R}^3$,
$F(u,v)=$(u sin$\alpha$ cos$v$, u sin$\alpha$ sin$v$, u cos$\alpha$),
$(u,v)\in U= \{(u,v)\in \mathbb{R}^2; u>0\}$, $\alpha=$const.
Is F a local isometry?
My attempt/ problem:
I believe the answer should be yes (since plane and cone are indeed locally isometric, and being isometric should not be dependent on parametrization.) To show that two surfaces are isometric, we need to show they have the same first fundamental forms. I do not know how to parametrize plane to get the same E, F, G as what we get for the above parametrization of cone, which is E= 1, F=0, G= $u^2sin^2\alpha$.