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

 

 

Here further references about surfaces:

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