This question is based on a construction in this lecture from the WE-Heraeus International Winter School on Gravity and Light 2015. Most of the question is context, just the last paragraph is really the point.
We work in the (quadrant I) stationary patch $\mathcal{U}$ of the Schwarzschild spacetime $M$ which can be described by the Schwarzschild coordinate chart $x: \mathcal{U} \to \mathbb{R}^4$ restricted to $r \in (2m,\infty)$. The metric is famously then $$g = -(1-2M/r) dt^2 + \frac{dr^2}{1-2M/r}+r^2d\Omega^2$$ Furthermore we restrict to the submanifold defined by (colatitude) $\theta = \frac{\pi}{2}$ giving a closed embedded three-manifold with induced metric $$\tilde{g} = -(1-2M/r) dt^2 + \frac{dr^2}{1-2M/r}+r^2d\phi^2$$ Side question: this manifold has topology $\mathbb{R}^2 \times \mathbb{R}$ as a submanifold of $\mathcal{U}$, correct?
Main question: the lecturer then lets $\tilde{g} = 0$ (he is thinking in terms of line elements), with the hope of analyzing lightlike curves, and rearranging the induced metric into the "optical metric"
$$dt^2 = \frac{dr^2}{(1-2M/r)^2} + \frac{r^2 d\phi^2}{1-2M/r}$$
thus constructing a Riemannian 2-manifold, a surface, on which the geodesics (by a version of Fermat's theorem) will correspond to lightlike geodesics in the $\theta = \pi/2$ plane of $\mathcal{U}$. How do we make this construction mathematically precise?
My idea: we look at the condition for a curve to be lightlike in the submanifold (and hence in $\mathcal{U}$): $$0 = (\frac{dt}{d\lambda})^2 g_{tt}+(\frac{dr}{d\lambda})^2g_{rr} + (\frac{d\phi}{d\lambda})^2 g_{\phi \phi}$$ and choose as the curve parameter Schwarzschild time $t$. And so $$0 = -(1-2M/r)+(\frac{dr}{dt})^2(1-2M/r)^{-1} + (\frac{d\phi}{dt})^2 r^2$$ in other words $$1 = (1-2M/r)^{-2}(\frac{dr}{dt})^2 + r^2(1-2M/r)^{-1}(\frac{d\phi}{dt})^2$$ which agrees with the optical metric with a suspicious multiplication by $dt^2$. How do the curves which satisfy this equation actually comprise a Riemannian manifold with the "optical metric"? Are they a folitation of the submanifold?