Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$?

Question

Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$?

I think that it is, because it can be parameterized by $f(x) = (\frac{1}{2},\sqrt{\frac{3}{4}} \cos x, \sqrt{\frac{3}{4}} \sin x)$ for $0 < x < 2\pi$.

Is this correct?

Answer 1

Let $$f:\Bbb R^3\to\Bbb R^2,\quad f(x,y,z)=(x^2+y^2+z^2-1,x-1/2).$$ Then, your space is $f^{-1}(0)$. Now, $$df=\begin{pmatrix} 2x & 2y & 2z \\ 1 & 0 & 0 \end{pmatrix} $$ which has full rank for all $(x,y,z)\in f^{-1}(0)$, so yes it is a manifold.