📂Geometry

Fundamental Theorem of Curved Surfaces

Theorem¹

For an open set $U \subset \mathbb{R}^{2}$, suppose any two points within $U$ are connected by a curve within $U$. Also, let the function $L_{ij}, g_{ij} : U \to \mathbb{R}\ (i,j = 1,2)$ be differentiable and have the following properties:

1. $L_{12} = L_{21}$, $g_{12} = g_{21}$, $g_{11}, g_{22} > 0$, and $g_{11}g_{22} - (g_{12})^{2} > 0$
2. Assume that $L_{ij}, g_{ij}$ satisfies the Gauss equation and the Codazzi-Mainardi equation.

$$\dfrac{\partial \Gamma_{ik}^{l}}{\partial u^{j}} - \dfrac{\partial \Gamma_{ij}^{l}}{\partial u^{k}} + \sum_{p} \left( \Gamma_{ik}^{p} \Gamma_{pj}^{l} - \Gamma_{ij}^{p}\Gamma_{pk}^{l}\right) = L_{ik}L_{j}^{l} - L_{ij}L_{k}^{l}$$

$$\dfrac{\partial L_{ij}}{\partial u^{k}_{}} - \dfrac{\partial L_{ik}}{\partial u^{j}} = \sum\limits_{l} \left( \Gamma_{ik}^{l}L_{lj} - \Gamma_{ij}^{l}L_{lk} \right)$$

Then, $\Gamma_{ij}^{k} = \dfrac{1}{2} \sum \limits_{l=1}^{2} g^{lk} \left( \dfrac{\partial g_{lj}}{\partial u_{i}} - \dfrac{\partial g_{ij}}{\partial u_{l}} + \dfrac{\partial g_{il}}{\partial u_{j}} \right)$ holds.

Subsequently, for $p \in U$, there exists a unique open set $V$ such that '\there exists a coordinate patch mapping $\mathbf{x} : V \to \mathbb{R}^{3}$ having $p \in V \subset U$ and $g_{ij}$ and $L_{ij}$ as coefficients of the first fundamental form and the second fundamental form'.

Explanation

The essence of the Fundamental Theorem of Curves was that ‘curves are uniquely determined by curvature and torsion’ and ‘for a differentiable $\overline{\kappa} >0$ and continuous $\overline{\tau}$, there exists a curve having them as its curvature and torsion’.

Similarly, the Fundamental Theorem of Surfaces states that ‘surfaces are uniquely determined by the Gauss equation and the Codazzi-Mainardi equation’.