In the book Differential Geometry by Lipschutz, the Gauss-Weingarten equations are written as follows:
On a patch $\mathbf{x}=\mathbf{x}(u^1,u^2)$ on a surface of class $\geq2$ the vectors $\mathbf{x}_i=\frac{\partial\mathbf{x}}{\partial u^i}$, $\mathbf{x}_{ij}=\frac{\partial^2\mathbf{x}}{\partial u^i \partial u^j}$ and $\mathbf{N}_i=\frac{\partial\mathbf{N}}{\partial u^i}$, with $i,j=1,2$, satisfy:
$$\mathbf{x}_{1 1}=\Gamma_{11}^{1} \mathbf{x}_{1}+\Gamma_{11}^{2} \mathbf{x}_{2}+b_{11} \mathbf{N} $$
$$\mathbf{x}_{1 2}=\Gamma_{12}^{1} \mathbf{x}_{1}+\Gamma_{12}^{2} \mathbf{x}_{2}+b_{12} \mathbf{N} $$
$$\mathbf{x}_{2 2}=\Gamma_{22}^{1} \mathbf{x}_{1}+\Gamma_{22}^{2} \mathbf{x}_{2}+b_{22} \mathbf{N} $$
$$\mathbf{N}_{1}=\beta_{1}^{1} \mathbf{x}_{1}+\beta_{1}^{2} \mathbf{x}_{2}$$
$$\mathbf{N}_{2}=\beta_{2}^{1} \mathbf{x}_{1}+\beta_{2}^{2} \mathbf{x}_{2}$$
With $b_{ij}$ the second-order fundamental coefficients.
Using the Einstein summation convention, with $i,j,\alpha=1,2$, I see that it can be expressed more compatly as:
$$\mathbf{x}_{i j}=\Gamma_{i j}^{\alpha} \mathbf{x}_{\alpha}+b_{i j} \mathbf{N}$$
$$\mathbf{N}_i=\beta^\alpha _i \mathbb{x}_\alpha$$
However, in the book the coefficients $\Gamma_{i j}^{\alpha}$ and $\beta^\alpha _i$ are written in a way that gives you a headache:
$$\Gamma_{11}^1=\frac{g_{22}g_{11,1}-2g_{12}g_{12,1}+g_{12}g_{11,2}}{2\left(g_{11}g_{22}-g_{12}^2\right)},\quad \Gamma_{12}^1=\frac{g_{22}g_{11,2}-g_{12}g_{22,1}}{2\left(g_{11}g_{22}-g_{12}^2\right)}, \quad \Gamma_{22}^1=\frac{2g_{22}g_{12,2}-g_{22}g_{22,1}-g_{12}g_{22,2}}{2\left(g_{11}g_{22}-g_{12}^2\right)}$$
$$\Gamma_{11}^2=\frac{2g_{11}g_{12,1}-g_{11}g_{11,2}+g_{12}g_{11,1}}{2\left(g_{11}g_{22}-g_{12}^2\right)}, \quad \Gamma_{12}^2=\frac{g_{11}g_{22,1}-g_{12}g_{11,2}}{2\left(g_{11}g_{22}-g_{12}^2\right)}, \quad \Gamma_{22}^2=\frac{g_{11}g_{22,2}-2g_{12}g_{12,2}+g_{12}g_{22,1}}{2\left(g_{11}g_{22}-g_{12}^2\right)}$$
$$\beta_1^1=\frac{b_{12}g_{12}-b_{11}g_{22}}{g_{11}g_{22}-g_{12}^2},\quad \beta_1^2=\frac{b_{11}g_{12}-b_{12}g_{11}}{g_{11}g_{22}-g_{12}^2},\quad \beta_2^1=\frac{b_{22}g_{12}-b_{12}g_{22}}{g_{11}g_{22}-g_{12}^2},\quad \beta_2^2=\frac{b_{12}g_{12}-b_{22}g_{11}}{g_{11}g_{22}-g_{12}^2}$$
Where $\{g_{ij}\}$ and $\{b_{ij}\}$ are the first and second-order fundamental coefficients, respectively, and where $g_{ij,k}=\frac{\partial g_{ij}}{\partial u^k}$.
Is there a compact way to express the coefficients $\Gamma_{i j}^{\alpha}$ and $\beta^\alpha _i$ in terms of the first and second-order fundamental coefficients, $g_{ij}$ and $b_{ij}$?
Best Answer
I answer myself: the Christoffel symbols can be expressed in terms of the first fundamental coefficients as
$$\Gamma_{i j}^{\alpha} = \frac{1}{2} g^{\alpha \gamma} \left[g_{j \gamma,i}+{g_{\gamma i,j}} - {g_{i j,\gamma}} \right] \tag{1}$$
with $i,j,\alpha, \gamma=1,2$, and the coefficients $\beta^\alpha _i$ can be expressed in terms of the second fundamental coefficients as
$$\beta^\alpha _i = - g^{\alpha \gamma}b_{i \gamma} \tag{2}$$
So the Gauss-Weingarten equations and their coefficients can be written compactly as
$$\mathbf{x}_{i j}=\Gamma_{i j}^{\alpha} \mathbf{x}_{\alpha}+b_{i j} \mathbf{N}$$ $$\mathbf{N}_i=- g^{\alpha \gamma}b_{i \gamma} \mathbb{x}_\alpha$$
With the coefficients $\Gamma_{i j}^{\alpha}$ given by $(1)$.