Fig. 2.

Scalar curvature is accurately estimated by using the Second Fundamental Form and the Gauss–Codazzi equation. (A) A hypothetical manifold (shown in gray) from which data points are sampled (shown as colored dots). The manifold at any given point $p$ (shown in red) can be decomposed into a tangent space $T_M\left(p\right)$ (the cyan plane) and a normal space $N_M\left(p\right)$ (the cyan line). Points in the neighborhood around $p$ (shown in green) can be expressed in terms of orthonormal bases for $T_M\left(p\right)$ and $N_M\left(p\right)$ (see B). (B) The set of points in the neighborhood of $p$ (shown as green dots in A) are represented here in local tangent $\left(t_1,t_2\right)$ and normal $\left(n_1\right)$ coordinates, corresponding to orthonormal bases for $T_M\left(p\right)$ and $N_M\left(p\right)$, respectively. Coloring corresponds to magnitude in the normal direction. The normal coordinates $\left(n_1\right)$ can be locally approximated as a quadratic function (the translucent surface) of the tangent coordinates $\left(t_1,t_2\right)$, according to the Second Fundamental Form, $h_{ij}^k$. (C) Scalar curvatures computed by using the extrinsic approach for $N=10^4$ points uniformly sampled from the two-dimensional hollow unit sphere, ${\mathcal{S}}^2$. The true value is two at all points on the manifold (SI Appendix, Supporting Methods, section D.1.1). (D) Scalar curvatures ($S$) computed in C are plotted against their associated SEs (${\sigma }_S$). Points enclosed by the red lines have a 95% CI, computed as $S\pm 2{\sigma }_S$, containing the true value of two. (E) As in C, but for $N=10^4$ points uniformly sampled from a one-sheet hyperboloid, $H_2^2$, which is also a two-dimensional manifold. Due to the radial symmetry of the manifold, scalar curvature only varies only along the $z$ direction (SI Appendix, Supporting Methods, section D.1.2). (F) Scalar curvatures (black) computed in E with their associated 95% CIs (shown in gray) plotted as a function of the z coordinates of the data points. The true value is shown as a dashed red line. (G) As in C, but for $N=10^4$ points uniformly sampled from a two-dimensional ring torus, $T^2$. $T^2$ is constructed by revolving a circle parameterized by $\theta$, oriented perpendicular to the $xy$ plane, through an angle $\varphi$ around the $z$ axis. The scalar curvature only depends on the value of $\theta$ (SI Appendix, Supporting Methods, section D.1.3). (H) Scalar curvatures computed in G with their associated 95% CIs plotted as a function of the $\theta$ values of the data points. Colors are as in F. (I) Distribution of computed scalar curvatures for $N=10^4$ points uniformly sampled from the $d$-dimensional unit hypersphere, ${\mathcal{S}}^d$, for $d=\mathrm{2,3,5,7}$. As with ${\mathcal{S}}^2$, these manifolds are isotropic and have constant scalar curvature. The true values are shown as dashed red lines (SI Appendix, Supporting Methods, section D.1.1).