Reducing the Dimensionality of the Sphere in terms of the Lie Algebra

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'?

• It's not quite clear to me what do want. Are you looking for an reasonable, explicit subspace $U \subset SO(n + 1)$ whose image under the canonical projection $SO(n + 1) \to SO(n + 1) / SO(n)$ can be identified with a (great) hypersphere $S^{n - 1}$ in $S^n$? – Travis Apr 11 at 18:10