Prove that the below two sets are submanifolds of $\mathbb{R}^n$.

Question

I was asked to prove that the sets $$X_1=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4: \sum x_i^2=1, x_1x_2x_3=1\}$$ And $$X_\epsilon=\{(z_1,z_2,z_3,z_4,z_5)\in \mathbb{C}^5: z_1^5+z_2^3+z_3^2+z_4^2+z_5^2=0,\sum |z_i|^2=\epsilon$$ Are indeed smooth submanifolds of $\mathbb{R}^4$ and $\mathbb{C}^5$ respectively. My general idea was to show that they are level sets of a submersion, but it’s still difficult to compute the tangent vectors to the two spheres (especially when the dimension is high). Is there an easier, quicker, by-definition way to do this?

Answer 1

Yes: in fact one can, instead of considering the conditions as two maps $$f_1=\sum x_i^2,f_2=x_1x_2x_3-1,$$ We may consider the single function $$f:\mathbb{R}^4 \to \mathbb{R}^2,f(x)=(f_1,f_2)$$ Then, compute and show that the Jacobian has full rank in the subset.