Metric signature: $\sigma$

Normal vector:
$n_a = (1, 0)$

Projection tensor:
$\gamma_{ab} = g_{ab} - \sigma n_a n_b$

Extrinsic curvature:
$K_{ab} = -{\gamma_a}^p {\gamma_b}^q \nabla_p n_q$
$K_{ab} = -\alpha {\Gamma^\perp}_{ab}$
Where ${\Gamma^\perp}_{ab}$ is the connection in the coordinate perpindicular to the horizontal components.

2D Riemann Curvature:
$R_{abcd} = \frac{1}{2} R (g_{ac} g_{bd} - g_{ad} g_{bc})$

Gauss's equation:
$R^\perp_{abcd} - \sigma (K_{ac} K_{bd} - K_{ad} K_{bc}) = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}$

Codazzi's equation:
$\sigma (\nabla^\perp_b K_{ac} - \nabla^\perp_a K_{bc}) = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r n^s R_{pqrs}$

---

Polar:
$g_{rr} = 1$; $g_{\phi\phi} = r^2$;
$R_{r\phi r\phi} = 0$

Sphere Surface:
$g_{\theta\theta} = r^2$; $g_{\phi\phi} = r^2 sin(\theta)^2$
$R_{\theta\phi\theta\phi} = r^2 sin(\theta)^2$

Sphere:
$g_{rr} = 1$; $g_{\theta\theta} = r^2$; $g_{\phi\phi} = r^2 sin(\theta)^2$
${\Gamma^r}_{\theta\theta} = -r$; ${\Gamma^r}_{\phi\phi} = -r sin(\theta)^2$
$R_{abcd} = 0$

Relation between Sphere Surface and Sphere:
'lapse': $\alpha = \sqrt{g_{rr}} = 1$
$R^\perp_{\theta\phi\theta\phi} - K_{\theta\theta} K_{\phi \phi} + (K_{\theta\phi})^2 = {\gamma_\theta}^a {\gamma_\phi}^b {\gamma_\theta}^c {\gamma_\phi}^d R_{abcd}$
$r^2 sin(\theta)^2 - K_{\theta\theta} K_{\phi\phi} + (K_{\theta\phi})^2 = 0$
$r^2 sin(\theta)^2 - (-\alpha {\Gamma^r}_{\theta\theta}) (-\alpha {\Gamma^r}_{\phi\phi}) + (-\alpha {\Gamma^r}_{\theta\phi})^2 = 0$
$r^2 sin(\theta)^2 - r \cdot r sin(\theta)^2 = 0$

Relation between Polar and Sphere:
$R^\perp_{r\phi r\phi} - K_{rr} K_{\phi \phi} + (K_{r\phi})^2 = {\gamma_r}^a {\gamma_\phi}^b {\gamma_r}^c {\gamma_\phi}^d R_{abcd}$
$0 - 0 \cdot r sin(\theta)^2 + 0 \cdot 0 = 0$