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Rebesques · #2 June 5th, 2009, 01:30 PM

Ok... some hints.

a) Show $|| f_u\times f_v||\neq 0$ for all $(u,v)$ . A basis for $T_pf$ is $\{f_u(p),f_v(p)\}$ .

To show that $e_1\notin T_pf \ \forall p$ , express $e_1$ as a linear combination of the basis vectors and show this cannot be.

b) Compute $\frac{\partial(u,v)}{\partial(s,t)}(s,t)$ and show it is nonzero for all $(s,t)$ .

c) Check that $(P_s\times P_t)(q)=(0,0,1)$ , so that the tangent space $T_qP$ contains $e_1$ .