Theorem: the Gaussian curvature K of a regular surface S \subset \mathbb{R}^3 is an intrinsic quantity. It depends exclusively upon the coefficients of the first fundamental form g_{ij} and their partial derivatives of the first and second order.
Proof: let S be a regular surface parametrized by local coordinates (u^1, u^2) with a position vector \mathbf{x}(u^1, u^2). The partial derivatives \mathbf{x}_i = \partial_i \mathbf{x} = \frac{\partial \mathbf{x}}{\partial u^i} form a basis for the tangent plane at each point. Let \mathbf{n} be the unit normal vector to the surface. The set \{\mathbf{x}_1, \mathbf{x}_2, \mathbf{n}\} forms a local basis for the ambient space \mathbb{R}^3.
The derivatives of these basis vectors are described by the Gauss-Weingarten equations. The second derivatives of the position vector, \mathbf{x}_{ij} = \partial_j \partial_i \mathbf{x}, are decomposed into tangential and normal components via the Gauss formula:
\mathbf{x}_{ij} = \Gamma^k_{ij} \mathbf{x}_k + L_{ij} \mathbf{n}
Here, summation over the repeated index k \in \{1,2\} is implied. The coefficients \Gamma^k_{ij} are the Christoffel symbols of the second kind. They are determined entirely by the metric tensor components g_{ij} = \mathbf{x}_i \cdot \mathbf{x}_j and their first partial derivatives:
\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right)
The coefficients L_{ij} = \mathbf{x}_{ij} \cdot \mathbf{n} define the second fundamental form.
The derivatives of the normal vector are given by the Weingarten equations:
\mathbf{n}_j = \partial_j \mathbf{n} = -L_j^k \mathbf{x}_k
where L_j^k = g^{kl} L_{jl} are the components of the shape operator.
The ambient space \mathbb{R}^3 is Euclidean, so its curvature is zero. This implies that mixed partial derivatives of the vector field \mathbf{x} commute: \partial_k (\partial_j \mathbf{x}_i) = \partial_j (\partial_k \mathbf{x}_i). This condition can be written as \partial_k \mathbf{x}_{ij} - \partial_j \mathbf{x}_{ik} = \mathbf{0}. We expand the term \partial_k \mathbf{x}_{ij} using the product rule and substitute the Gauss-Weingarten equations:
\begin{aligned} \partial_k \mathbf{x}_{ij} &= \partial_k (\Gamma^l_{ij} \mathbf{x}_l + L_{ij} \mathbf{n}) \\ &= (\partial_k \Gamma^l_{ij}) \mathbf{x}_l + \Gamma^l_{ij} \mathbf{x}_{lk} + (\partial_k L_{ij}) \mathbf{n} + L_{ij} \mathbf{n}_k \\ &= (\partial_k \Gamma^l_{ij}) \mathbf{x}_l + \Gamma^l_{ij} (\Gamma^m_{lk} \mathbf{x}_m + L_{lk} \mathbf{n}) + (\partial_k L_{ij}) \mathbf{n} - L_{ij} (L_k^m \mathbf{x}_m) \end{aligned}
Collecting the components tangential to S (the terms multiplying \mathbf{x}_m) and normal to S (the terms multiplying \mathbf{n}) gives:
\partial_k \mathbf{x}_{ij} = \left[ (\partial_k \Gamma^m_{ij}) + \Gamma^l_{ij} \Gamma^m_{lk} - L_{ij} L_k^m \right] \mathbf{x}_m + \left[ \partial_k L_{ij} + \Gamma^l_{ij} L_{lk} \right] \mathbf{n}
The vector equation \partial_k \mathbf{x}_{ij} - \partial_j \mathbf{x}_{ik} = \mathbf{0} must hold for both tangential and normal components separately. The vanishing of the tangential component provides the relation:
\left[ (\partial_k \Gamma^m_{ij}) + \Gamma^l_{ij} \Gamma^m_{lk} - L_{ij} L_k^m \right] - \left[ (\partial_j \Gamma^m_{ik}) + \Gamma^l_{ik} \Gamma^m_{lj} - L_{ik} L_j^m \right] = 0
By rearranging terms, we separate the quantities derived from the first fundamental form from those derived from the second:
\partial_j \Gamma^m_{ik} - \partial_k \Gamma^m_{ij} + \Gamma^l_{ik} \Gamma^m_{lj} - \Gamma^l_{ij} \Gamma^m_{lk} = L_{ik} L_j^m - L_{ij} L_k^m
The left-hand side defines the component R^m_{ikj} of the Riemann curvature tensor. Lowering the first index by contracting with the metric g_{mq} gives the fully covariant form of the Riemann tensor, R_{qikj} = g_{mq}R^m_{ikj}. This operation yields the Gauss equation:
R_{qikj} = L_{ik} L_{qj} - L_{ij} L_{qk}
The Gaussian curvature K is defined for a two-dimensional manifold by K = \frac{R_{1212}}{g}, where g = \det(g_{ij}). From the Gauss equation, we obtain:
R_{1212} = L_{11}L_{22} - L_{12}L_{21} = \det(L_{ij})
This leads to the computational formula K = \det(L_{ij}) / \det(g_{ij}).
The substance of the theorem, however, lies in the fact that the Riemann tensor R_{qikj}, by its very definition, is constructed exclusively from the Christoffel symbols and their derivatives.
The Christoffel symbols are, in turn, functions only of the metric tensor g_{ij} and its first derivatives.
Therefore, R_{qikj} depends only on the metric and its first and second derivatives.
As K is a scalar obtained from the Riemann tensor and the metric, it must be an intrinsic property of the surface, calculable without reference to the specific embedding in \mathbb{R}^3.