Curve Immersed in Surface

This is a note for Nadir’s work, but I have not fully understood it.

Overview

From the perspective of a growth model, for growth along the $t$ direction: $ \frac{\partial \mathbf{X}}{\partial t}=\mathrm{U}\mathbf{n} $

Define the metric of the surface:

$ g_{11}=\frac{\partial \mathbf{X}}{\partial \sigma}\cdot\frac{\partial \mathbf{X}}{\partial \sigma}=\frac{\partial\mathbf{X}}{\partial s}\cdot\frac{\partial\mathbf{X}}{\partial s}*(\frac{\partial s}{\partial \sigma})^2=(\frac{\partial s}{\partial \sigma})^2 $

$ g_{22}=\frac{\partial \mathbf{X}}{\partial t}\cdot\frac{\partial \mathbf{X}}{\partial t}=\mathrm{U}^2 $

$ g_{12}=0 $

Derivative of the metric

$ \frac{\partial g}{\partial \sigma}=2g\sqrt{g}\frac{\partial \mathbf{X}_s}{\partial \sigma}\mathbf{X}_s=0 $

$ \frac{\partial \sqrt{g}}{\partial t}=2g\mathrm{U}\frac{\partial \mathbf{n}}{\partial s}\mathbf{T}=-\sqrt{g}\kappa _g\mathrm{U} $

Transformation relations of the curve frame

${\mathrm{T},\mathrm{n},\mathrm{N}}$ describes the motion of a curve on the surface. Along the ${s,t}$ directions, due to the orthonormality of the moving frame, the corresponding equations of motion must be satisfied.

s-direction

In the $s$ direction, the Darboux frame satisfies: \(\partial _s\left( \begin{array}{c} \mathbf{T}\\ \mathbf{n}\\ \mathbf{N}\\ \end{array} \right) =\left( \begin{matrix} 0& \kappa _g& \kappa _N\\ -\kappa _g& 0& \tau _g\\ -\kappa _N& -\tau _g& 0\\ \end{matrix} \right) \left( \begin{array}{c} \mathbf{T}\\ \mathbf{n}\\ \mathbf{N}\\ \end{array} \right)\)

$ \kappa_n=\frac{\partial^2\mathbf{X}}{\partial s^2}\cdot\mathbf{N} $

$ \kappa_g=\frac{\partial^2\mathbf{X}}{\partial s^2}\cdot\mathbf{n} $

t-direction

First, derive the commutation relation of partial derivatives: $ \partial _t\partial _s=\partial _t\left( \frac{1}{\sqrt{g}}\partial _{\sigma} \right) =\partial _s\partial _t+\partial _t\left( \frac{1}{\sqrt{g}} \right) \partial _{\sigma}=\partial _s\partial _t+\kappa _g\mathrm{U}\partial _s $ In the $t$ direction, the Darboux frame satisfies: \(\partial _t\left( \begin{array}{c} \mathbf{T}\\ \mathbf{n}\\ \mathbf{N}\\ \end{array} \right) =\left( \begin{matrix} 0& \frac{\partial \mathrm{U}}{\partial s}& \mathrm{U}\tau _g\\ -\frac{\partial \mathrm{U}}{\partial s}& 0& \mathrm{U}\kappa _{N,2}\\ -\mathrm{U}\tau _g& -\mathrm{U}\kappa _{N,2}& 0\\ \end{matrix} \right) \left( \begin{array}{c} \mathbf{T}\\ \mathbf{n}\\ \mathbf{N}\\ \end{array} \right)\)

Surface constitutive conditions

First, obtain the coefficients of the first and second fundamental forms: $ \mathrm{E}=(\frac{\partial s}{\partial \sigma})^2=g;\mathrm{F}=0;\mathrm{G}=\mathrm{U}^2 $

$ \mathrm{L}=\frac{\partial ^2\mathbf{X}}{\partial \sigma ^2}\cdot \mathbf{N}=\sqrt{g}\frac{\partial}{\partial s}\left( \mathbf{T}\sqrt{g} \right) \cdot \mathbf{N}=g\kappa _N $

$ \mathrm{M}=\frac{\partial ^2\mathbf{X}}{\partial \sigma \partial t}\cdot \mathbf{N}=\frac{\partial s}{\partial \sigma}\frac{\partial}{\partial s}\left( \mathrm{U}\mathbf{n} \right) \cdot \mathbf{N}=\sqrt{g}\mathrm{U}\tau _g $

$ \mathrm{N}=\frac{\partial ^2\mathbf{X}}{\partial t^2}\cdot \mathbf{N}=\frac{\partial}{\partial t}\left( \mathrm{U}\mathbf{n} \right) \cdot \mathbf{N}=\mathrm{U}\frac{\partial \mathbf{n}}{\partial t}\cdot \mathbf{N}=\mathrm{U}^2\frac{\partial \mathbf{n}}{\mathrm{U}\partial t}\cdot \mathbf{N}=\mathrm{U}^2\kappa _{N,2} $

The Christoffel symbols of the surface computed from Eqs. (8) ~ (11) are: $ \Gamma _{11}^{1}=\frac{\mathrm{E}_u}{2\mathrm{E}}=\frac{\partial \sqrt{g}}{\partial s}; \Gamma _{12}^{1}=\frac{\mathrm{E}_v}{2\mathrm{E}}=-\mathrm{U}\kappa _g; \Gamma _{22}^{1}=-\frac{\mathrm{G}_u}{2\mathrm{E}}=-\frac{\mathrm{U}}{\sqrt{g}}\frac{\partial \mathrm{U}}{\partial s} $

$ \Gamma _{22}^{2}=\frac{\mathrm{G}_v}{2\mathrm{G}}=\frac{1}{\mathrm{U}}\frac{\partial \mathrm{U}}{\partial t}; \Gamma _{12}^{2}=\frac{\mathrm{G}_u}{2\mathrm{G}}=\frac{\sqrt{g}}{\mathrm{U}}\frac{\partial \mathrm{U}}{\partial s}; \Gamma _{11}^{2}=-\frac{\mathrm{E}_v}{2\mathrm{G}}=\frac{g}{\mathrm{U}}\kappa _g $

Codazzi-Mainardi equations

$ \Gamma _{12}^{1}\mathrm{L}+\left( \Gamma _{12}^{2}-\Gamma _{11}^{1} \right) \mathrm{M}-\Gamma _{11}^{2}\mathrm{N}=\mathrm{L}_v-\mathrm{M}_u $

$ \Gamma _{22}^{1}\mathrm{L}+\left( \Gamma _{22}^{2}-\Gamma _{21}^{1} \right) \mathrm{M}-\Gamma _{21}^{2}\mathrm{N}=\mathrm{M}_v-\mathrm{N}_u $

Substituting into the Mainardi equations gives: $ -\mathrm{Ug}\kappa _N\kappa _g+\left( \frac{\sqrt{g}}{\mathrm{U}}\frac{\partial \mathrm{U}}{\partial s}-\frac{\partial \sqrt{g}}{\partial s} \right) \sqrt{\mathrm{g}}\mathrm{U}\tau _g-\frac{g}{\mathrm{U}}\kappa _g\mathrm{U}^2\kappa _{N,2}=\frac{\partial \mathrm{g}\kappa _N}{\partial t}-\frac{\partial \left( \sqrt{\mathrm{g}}\mathrm{U}\tau _g \right)}{\partial \sigma} $

$ -\frac{\mathrm{U}}{\sqrt{g}}\frac{\partial \mathrm{U}}{\partial s}\mathrm{g}\kappa _N+\left( \mathrm{U}\kappa _g+\frac{1}{\mathrm{U}}\frac{\partial \mathrm{U}}{\partial t} \right) \sqrt{\mathrm{g}}\mathrm{U}\tau _g-\frac{\sqrt{g}}{\mathrm{U}}\frac{\partial \mathrm{U}}{\partial s}\mathrm{U}^2\kappa _{N,2}=\frac{\partial \left( \sqrt{\mathrm{g}}\mathrm{U}\tau _g \right)}{\partial t}-\frac{\partial \left( \mathrm{U}^2\kappa _{N,2} \right)}{\partial \sigma} $

Simplifying: $ \frac{\partial \kappa _N}{\partial t}=\frac{\partial \left( \mathrm{U}\tau _g \right)}{\partial s}+\frac{\partial \mathrm{U}}{\partial s}\tau _g+\mathrm{U}\kappa _g\left( \kappa _N-\kappa _{N,2} \right) $

$ \frac{\partial \tau _g}{\partial t}=\frac{\partial \left( \mathrm{U}\kappa _{N,2} \right)}{\partial s}-\frac{\partial \mathrm{U}}{\partial s}\kappa _N+2\mathrm{U}\kappa _g\tau _g $

Gauss-Bonnet theorem