The geometry of unit vectors that have specific angle with a given vector It is easy to see that for $S^2$ this space is nothing but a circle that is the intersection of a cone with aperture $2\alpha$ (where $\alpha$ is the predifined specific angle), and $S^2$. My question is that is this observation extendable to higher dimensions? Is it true that for any given vector $u\in\mathbb{R}^n$ and the unit vectors in $\mathbb{R}^n$, the geometry of the points over this unit sphere that have a specific angle with $u$ is $S^{n-1}$? Any hints to prove disprove it.
DISCLAIMER: This is not a homework but rather a self observation (as it might be guessed from its silliness(?!))

General spherical coordinates are given by
$$ \begin{align*}
x_1 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\sin{\phi_{n-2}}\sin{\phi_{n-1}} \\ 
x_2 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\sin{\phi_{n-2}}\cos{\phi_{n-1}} \\
x_3 &=r\sin{\phi_1}\sin{\phi_2}\dotsm\cos{\phi_{n-2}} \\
&\vdots\\
x_{n-1} &=r\sin{\phi_1}\cos{\phi_2}\\
x_n &= r \cos{\phi_1}
\end{align*}$$
Setting $r=1$ and dotting this with $$ (0,\dotsc,0,1) $$ gives $\cos{\phi_1}$, so if we hold that constant, we have a sphere (looking at the similarity of the coordinates after setting $\phi_1$ constant), and we can see, again by the definition, that it has radius $\lvert \sin{\phi_1} \rvert$ (since the first $n-1$ coordinates have common factor $\sin^2{\phi_1}$, and by analogy with $r$ in the full space).
