# [SOLVED] Second Fundamental Form

• Feb 3rd 2009, 05:57 AM
fobos3
[SOLVED] Second Fundamental Form
Hi there. I'm trying to understand the Second Fundamental Form but I'm stuck on something. Let:

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

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

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

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

I don't understand what is $\displaystyle L$ and $\displaystyle \Gamma$
• Feb 3rd 2009, 07:24 AM
Laurent
Quote:

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

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

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

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

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

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

This is just the decomposition of $\displaystyle \textbf{X}_{ij}$ in a basis. Indeed, $\displaystyle (\textbf{X}_1,\textbf{X}_2)$ is a basis of the tangent plane, while $\displaystyle \textbf{U}$ is orthogonal to that plane, so that $\displaystyle (\textbf{X}_1,\textbf{X}_2,\textbf{U})$ is a basis of $\displaystyle \mathbb{R}^3$ (which depends on $\displaystyle (u_1,u_2)$). Then for instance $\displaystyle \Gamma_{ij}^1$ is the $\displaystyle \textbf{X}_1$-component of $\displaystyle \textbf{X}_{i j}$.
• Feb 3rd 2009, 01:07 PM
fobos3
Quote:

Originally Posted by Laurent
This is just the decomposition of $\displaystyle \textbf{X}_{ij}$ in a basis. Indeed, $\displaystyle (\textbf{X}_1,\textbf{X}_2)$ is a basis of the tangent plane, while $\displaystyle \textbf{U}$ is orthogonal to that plane, so that $\displaystyle (\textbf{X}_1,\textbf{X}_2,\textbf{U})$ is a basis of $\displaystyle \mathbb{R}^3$ (which depends on $\displaystyle (u_1,u_2)$). Then for instance $\displaystyle \Gamma_{ij}^1$ is the $\displaystyle \textbf{X}_1$-component of $\displaystyle \textbf{X}_{i j}$.

Never mind. It's explained 3 chapters ahead. It's the Christoffel symbol.
• Feb 3rd 2009, 11:38 PM
Laurent
Quote:

Originally Posted by fobos3
Never mind.