The $n$-sphere can be written as an $(n-1)$-sphere fibered over an interval $$ ds^2_{\Omega_n} = d\theta^2 + \sin^2 \theta\;d\Omega_{n-1}^2. $$ In these coordinates, when we impose that we keep $\theta$ fixed at a certain value (e.g. $\theta = \pi/2$) the resulting space becomes effectively that of the $(n-1)$-sphere $$ ds^2_{\Omega_n} \Big|_{\theta=\pi/2} = ds^2_{\Omega_{(n-1)}} $$ I am trying to understand this statement from a group theoretical point of view.
The $n$-sphere is a symmetric space and is equivalent to the coset $$ S^n = \frac{SO(n+1)}{SO(n)}, $$ What would now be the procedure in terms of the algebra that is equivalent to `keeping $\theta$ fixed'?