1. [SOLVED] Second Fundamental Form

Hi there. I'm trying to understand the Second Fundamental Form but I'm stuck on something. Let:

$\textbf{X}=(x(u^1,u^2),y(u^1,u^2),z(u^1,u^2))$ is a parameterised form of a surface.

$\textbf{X}_i=\dfrac{\partial \textbf{X}}{\partial u^i}$

$\textbf{X}_{i j}=\dfrac{\partial \textbf{X}^2}{\partial u^i \partial u^j}$

It's written $\textbf{X}_{i j}=\Gamma_{ij}^{r}\textbf{X}_r +L_{ij}\textbf{U}$ where $\textbf{U}$ is the normal vector to the surface.

I don't understand what is $L$ and $\Gamma$

2. Originally Posted by fobos3
Hi there. I'm trying to understand the Second Fundamental Form but I'm stuck on something. Let:

$\textbf{X}=(x(u^1,u^2),y(u^1,u^2),z(u^1,u^2))$ is a parameterised form of a surface.

$\textbf{X}_i=\dfrac{\partial \textbf{X}}{\partial u^i}$

$\textbf{X}_{i j}=\dfrac{\partial \textbf{X}^2}{\partial u^i \partial u^j}$

It's written $\textbf{X}_{i j}=\Gamma_{ij}^{r}\textbf{X}_r +L_{ij}\textbf{U}$ where $\textbf{U}$ is the normal vector to the surface.

I don't understand what is $L$ and $\Gamma$
This is just the decomposition of $\textbf{X}_{ij}$ in a basis. Indeed, $(\textbf{X}_1,\textbf{X}_2)$ is a basis of the tangent plane, while $\textbf{U}$ is orthogonal to that plane, so that $(\textbf{X}_1,\textbf{X}_2,\textbf{U})$ is a basis of $\mathbb{R}^3$ (which depends on $(u_1,u_2)$). Then for instance $\Gamma_{ij}^1$ is the $\textbf{X}_1$-component of $\textbf{X}_{i j}$.

3. Originally Posted by Laurent
This is just the decomposition of $\textbf{X}_{ij}$ in a basis. Indeed, $(\textbf{X}_1,\textbf{X}_2)$ is a basis of the tangent plane, while $\textbf{U}$ is orthogonal to that plane, so that $(\textbf{X}_1,\textbf{X}_2,\textbf{U})$ is a basis of $\mathbb{R}^3$ (which depends on $(u_1,u_2)$). Then for instance $\Gamma_{ij}^1$ is the $\textbf{X}_1$-component of $\textbf{X}_{i j}$.
Never mind. It's explained 3 chapters ahead. It's the Christoffel symbol.

4. Originally Posted by fobos3
Never mind.