[SOLVED] Second Fundamental Form

**fobos3** · February 3rd 2009, 05:57 AM

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 $\Gamma$

**Laurent** · February 3rd 2009, 07:24 AM

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 $\textbf{X}_{i j}$ .

**fobos3** · February 3rd 2009, 01:07 PM

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.

**Laurent** · February 3rd 2009, 11:38 PM

Originally Posted by fobos3

Never mind.

Thanks for your consideration...

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