# surface curvature, geodesics and embeddings

• This is the GPS of a surface $\Sigma$: $\left(\!\!\begin{array}{c}v\\w\end{array}\!\!\right)\stackrel{\Phi}\hookrightarrow\left(\!\!\begin{array}{c}x(v,w)\\y(v,w)\\z(v,w)\end{array}\!\!\right)$
• And these are the base for the tangent space: $\Phi_{,v}=\left(\!\!\begin{array}{c}\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial v}\\\frac{\partial z}{\partial v}\end{array}\!\!\right)$, $\Phi_{,w}=\left(\!\!\begin{array}{c}\frac{\partial x}{\partial w}\\\frac{\partial y}{\partial w}\\\frac{\partial z}{\partial w}\end{array}\!\!\right)$
• This is the base of the orthogonal complement of the tangent space of the surface in $\mathbb{R}^3$:  $n=\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}$
• This is an useful identity: $g=||\Phi_{,v}\times\Phi_{,w}||^2=\langle\Phi_{,v},\Phi_{,v}\rangle\langle\Phi_{,w},\Phi_{,w}\rangle-\langle\Phi_{,v},\Phi_{,w}\rangle^2=g_{11}g_{22}-{g_{12}}^2$
• The next sequence of equalities deduce an alternative formula for the gaussian curvature in terms of triple products:

$k_{\rm{gauss}}(\Sigma)=\frac{\langle\Phi_{,vv},n\rangle\langle\Phi_{,ww},n\rangle-\langle\Phi_{,vw},n\rangle^2}{\langle\Phi_{,v},\Phi_{,v}\rangle\langle\Phi_{,w},\Phi_{,w}\rangle-\langle\Phi_{,v},\Phi_{,w}\rangle^2}$

$=\frac{\langle\Phi_{,vv},n\rangle\langle\Phi_{,ww},n\rangle-\langle\Phi_{,vw},n\rangle^2}{g}$

$=\frac{\langle\Phi_{,vv},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle\langle\Phi_{,ww},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle-\langle\Phi_{,vw},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle^2}{g}$

$=\frac{\langle\Phi_{,vv},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle\langle\Phi_{,ww},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle-\langle\Phi_{,vw},\frac{\Phi_{,v}\times\Phi_{,w}}{||\Phi_{,v}\times\Phi_{,w}||}\rangle^2}{g}$

$=\frac{\langle\Phi_{,vv},\frac{\Phi_{,v}\times\Phi_{,w}}{\sqrt{g}}\rangle\langle\Phi_{,ww},\frac{\Phi_{,v}\times\Phi_{,w}}{\sqrt{g}}\rangle-\langle\Phi_{,vw},\frac{\Phi_{,v}\times\Phi_{,w}}{\sqrt{g}}\rangle^2}{g}$

$=\frac{\langle\Phi_{,vv},\Phi_{,v}\times\Phi_{,w}\rangle\langle\Phi_{,ww},\Phi_{,v}\times\Phi_{,w}\rangle-\langle\Phi_{,vw},\Phi_{,v}\times\Phi_{,w}\rangle^2}{g^2}$

$k_{\rm{gauss}}(\Sigma)=\frac{\langle\Phi_{,vv},\Phi_{,v}\times\Phi_{,w}\rangle\langle\Phi_{,ww},\Phi_{,v}\times\Phi_{,w}\rangle-\langle\Phi_{,vw},\Phi_{,v}\times\Phi_{,w}\rangle^2}{(\langle\Phi_{,v},\Phi_{,v}\rangle\langle\Phi_{,w},\Phi_{,w}\rangle-\langle\Phi_{,v},\Phi_{,w}\rangle^2)^2}$

• Remember that for the triple product $\langle A,B\times C\rangle=\det\left(\begin{array}{ccc}\!\!A^1&B^1&C^1\\\!\!A^2&B^2&C^2\\\!\!A^3&B^3&C^3\end{array}\!\!\right)$

There’s a method to associate a “3-legs” frame (drei-bein) attached to a curve in a surface:

• a parameterization of a curve $C$  in a surface $\Sigma$ in the space $\mathbb{R}^3$ is a composition $\beta=\Phi\circ\alpha$ of maps  $\alpha:I\to\Omega\subset\mathbb{R}^2$ and $\Phi:\Omega\to\Sigma$, which can be regarded as an arco-parameterization
• so the velocity information (tangent information) is $T=\frac{d\beta}{ds}$ which in terms of the chain rule gives:

$T=\left(\!\!\begin{array}{c}dx/ds\\dy/ds\\dz/ds\end{array}\!\!\right)$

• then employing the standard covariant derivative:

$D_TT=\left(\begin{array}{ccc}\frac{\partial}{\partial x}(\frac{dx}{dt})&\frac{\partial}{\partial y}(\frac{dx}{dt})&\frac{\partial}{\partial z}(\frac{dx}{dt})\\\frac{\partial}{\partial x}(\frac{dy}{dt})&\frac{\partial}{\partial y}(\frac{dy}{dt})&\frac{\partial}{\partial z}(\frac{dy}{dt})\\\frac{\partial}{\partial x}(\frac{dz}{dt})&\frac{\partial}{\partial y}(\frac{dz}{dt})&\frac{\partial}{\partial z}(\frac{dz}{dt})\end{array}\right)\left(\begin{array}{c}\frac{dx}{dt}\\\frac{dy}{dt}\\\frac{dz}{dt} \end{array}\right)$

$=\left(\begin{array}{c}\frac{\partial}{\partial x}(\frac{dx}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dx}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dx}{dt})\frac{dz}{dt}\\\frac{\partial}{\partial x}(\frac{dy}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dy}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dy}{dt})\frac{dz}{dt}\\\frac{\partial}{\partial x}(\frac{dz}{dt})\frac{dx}{dt}+\frac{\partial}{\partial y}(\frac{dz}{dt})\frac{dy}{dt}+\frac{\partial}{\partial z}(\frac{dz}{dt})\frac{dz}{dt}\end{array}\right)$

$=\left(\!\!\begin{array}{c}d^2x/ds^2\\d^2y/ds^2\\d^2z/ds^2\end{array}\!\!\right)=T'=\beta''$

• Now, with $T$ and $N=\frac{\partial_1\times\partial_2}{||\partial_1\times\partial_2||}$ at each point along the curve, a third vector in constructed $V=N\times T$, then by ordering $\{T,V,N\}$ we can get

$\begin{array}{c}D_TT=\\D_TV=\\D_TN=\end{array}\begin{array}{c}gV+kN\\-gT+tN\\-kT-tV\end{array}$

since $\langle T,T\rangle=\langle V,V\rangle=\langle N,N\rangle=1$ and $\langle T,N\rangle=\langle N,V\rangle=\langle V,T\rangle=0$

• matricially those can be summarized as

$\left(\!\!\begin{array}{c}D_TT\\D_TV\\D_TN\end{array}\!\!\right)=\left(\begin{array}{ccc}0&g&k\\-g&0&t\\-k&-t&0\end{array}\right) \left(\!\!\begin{array}{c}T\\V\\N\end{array}\!\!\right)$

this relation is similar to the Serret-Frenet relation (check the animation therein wikipedia) of a curve in $\mathbb{R}^3$