Simbolos de Christoffel de una superficie en $\mathbb{R}^3$

Símbolos de Christoffel para una superficie S(u,v) = (X(u,v), Y(u,v), Z(u,v))

Los coeficientes o símbolos de Christoffel de una superficie S(u,v) = (X(u,v), Y(u,v), Z(u,v)) en $\mathbb{R}^3$ son funciones escalares que son deducidas a partir del siguiente sistema de ecuaciones respectivamente.

$$ {\large \begin{align*} \frac{\partial^2 S}{\partial u^2} &= \Gamma_{11}^{1} \frac{\partial S}{\partial u} + \Gamma_{11}^{2} \frac{\partial S}{\partial v} + Q^{1}(u,v) \hat{n} \\ \frac{\partial^2 S}{\partial u \partial v} &= \Gamma_{12}^{1} \frac{\partial S}{\partial u} + \Gamma_{12}^{2} \frac{\partial S}{\partial v} + Q^{2}(u,v) \hat{n} \\ \frac{\partial^2 S}{\partial v \partial u} &= \Gamma_{21}^{1} \frac{\partial S}{\partial u} + \Gamma_{21}^{2} \frac{\partial S}{\partial v} + Q^{3}(u,v)  \hat{n} \\ \frac{\partial^2 S}{\partial v^2} &= \Gamma_{22}^{1} \frac{\partial S}{\partial u} + \Gamma_{22}^{2} \frac{\partial S}{\partial v} + Q^{4}(u,v)  \hat{n} \end{align*}} $$

Los coeficientes ${Q^1(u,v)}, {Q^2(u,v)}, {Q^3(u,v)}$ y ${Q^4(u,v)}$ pueden obtenerse al realizar el producto escalar con el vector normal unitario a las ecuaciones del sistema de ecuaciones anterior. Por consiguiente 

$$ {\large \begin{align*} & \left\{ \begin{array}{lcc} \left< \frac{\partial^2 S}{\partial u^2}, \hat{n} \right> &= \left< \Gamma_{11}^{1} \frac{\partial S}{\partial u} + \Gamma_{11}^{2} \frac{\partial S}{\partial v} + Q^{1}(u,v) \hat{n}, \hat{n} \right>\\ \left< \frac{\partial^2 S}{\partial u \partial v}, \hat{n} \right>&= \left< \Gamma_{12}^{1} \frac{\partial S}{\partial u} + \Gamma_{12}^{2} \frac{\partial S}{\partial v} + Q^{2}(u,v) \hat{n}, \hat{n} \right> \\ \left< \frac{\partial^2 S}{\partial v \partial u}, \hat{n} \right>&=\left< \Gamma_{21}^{1} \frac{\partial S}{\partial u} + \Gamma_{21}^{2} \frac{\partial S}{\partial v} + Q^{3}(u,v)  \hat{n}, \hat{n} \right> \\ \left< \frac{\partial^2 S}{\partial v^2}, \hat{n} \right> &= \left< \Gamma_{22}^{1} \frac{\partial S}{\partial u} + \Gamma_{22}^{2} \frac{\partial S}{\partial v} + Q^{4}(u,v) \hat{n}, \hat{n} \right> \end{array} \right. \\ \\ & \left\{ \begin{array}{lcc} L &= Q^{1}(u,v) \\  M &= Q^{2}(u,v) \\  M &= Q^{3}(u,v) \\  N &= Q^{4}(u,v) \end{array} \right.\end{align*}}  $$

Donde L, M y N son los coeficientes de la segunda forma fundamental de la superficie S(u, v) respectivamente.

Ahora para calcular los valores de $\Gamma_{ij}^{k}$ se deducen a partir de las derivadas parciales de los coeficientes de la primera forma fundamental. 

• Coeficiente $E_{u}$

$$ {\large \begin{align*} E &= \left< \frac{\partial S}{\partial u}, \frac{\partial S}{\partial u} \right> \\ \frac{\partial E}{\partial u} &= 2\left<\frac{\partial^2 S}{\partial u^2} , \frac{\partial S}{\partial u} \right> \\ \frac{\partial E}{\partial u} &= 2\left< \Gamma_{11}^{1} \frac{\partial S}{\partial u} + \Gamma_{11}^{2} \frac{\partial S}{\partial v} + Q^{1}(u,v) \hat{n}, \frac{\partial S}{\partial u} \right> \\ \frac{1}{2}\frac{\partial E}{\partial u} &= E\Gamma_{11}^{1} + F\Gamma_{11}^{2}  \end{align*}} $$

• Coeficiente $E_{v}$

$$ {\large \begin{align*} E &= \left< \frac{\partial S}{\partial u}, \frac{\partial S}{\partial u} \right> \\ \frac{\partial E}{\partial v} &= 2\left<\frac{\partial^2 S}{\partial u \partial v } , \frac{\partial S}{\partial u} \right> \\ \frac{\partial E}{\partial v} &= 2\left<  \Gamma_{12}^{1} \frac{\partial S}{\partial u} + \Gamma_{12}^{2} \frac{\partial S}{\partial v} + Q^{2}(u,v) \hat{n}, \frac{\partial S}{\partial u} \right> \\ \frac{1}{2}\frac{\partial E}{\partial v} &= E\Gamma_{12}^{1} + F\Gamma_{12}^{2}  \end{align*}} $$

• Coeficiente $F_{u}$

$$ {\large \begin{align*} F &= \left< \frac{\partial S}{\partial u}, \frac{\partial S}{\partial v} \right> \\ \frac{\partial F}{\partial u} &= \left<\frac{\partial^2 S}{\partial u^2} , \frac{\partial S}{\partial v} \right> + \left<\frac{\partial S}{\partial u} , \frac{\partial^2 S}{\partial v \partial u} \right> \\ \frac{\partial F}{\partial u} &= \left<\frac{\partial^2 S}{\partial u^2} , \frac{\partial S}{\partial v} \right> + \left< \frac{\partial^2 S}{\partial u \partial v}, \frac{\partial S}{\partial u} \right> \\ \frac{\partial F}{\partial u} &= \left< \Gamma_{11}^{1} \frac{\partial S}{\partial u} + \Gamma_{11}^{2} \frac{\partial S}{\partial v} + Q^{1}(u,v) \hat{n} , \frac{\partial S}{\partial v} \right> + \frac{1}{2}\frac{\partial E}{\partial v} \\  \frac{\partial F}{\partial u} - \frac{1}{2}\frac{\partial E}{\partial v} &= F\Gamma_{11}^{1} + G\Gamma_{11}^{2} \end{align*}} $$

• Coeficiente $G_{v}$

$$ {\large \begin{align*} G &= \left< \frac{\partial S}{\partial v}, \frac{\partial S}{\partial v} \right> \\ \frac{\partial G}{\partial v} &= 2\left<\frac{\partial^2 S}{\partial v^2} , \frac{\partial S}{\partial v} \right> \\ \frac{\partial G}{\partial v} &= 2\left< \Gamma_{22}^{1} \frac{\partial S}{\partial v} + \Gamma_{22}^{2} \frac{\partial S}{\partial v} + Q^{4}(u,v)  \hat{n} , \frac{\partial S}{\partial v} \right> \\ \frac{1}{2}\frac{\partial G}{\partial v} &= F\Gamma_{22}^{1} + G\Gamma_{22}^{2}  \end{align*}} $$

• Coeficiente $G_{u}$

$$ {\large \begin{align*} G &= \left< \frac{\partial S}{\partial v}, \frac{\partial S}{\partial v} \right> \\ \frac{\partial G}{\partial u} &= 2\left<\frac{\partial^2 S}{\partial v \partial u} , \frac{\partial S}{\partial v} \right> \\ \frac{\partial G}{\partial u} &= 2\left< \Gamma_{21}^{1} \frac{\partial S}{\partial u} + \Gamma_{21}^{2} \frac{\partial S}{\partial v} + Q^{3}(u,v)  \hat{n} , \frac{\partial S}{\partial v} \right> \\ \frac{1}{2}\frac{\partial G}{\partial u} &= F\Gamma_{21}^{1} + G\Gamma_{21}^{2}  \end{align*}} $$

• Coeficiente $F_{v}$

$$ {\large \begin{align*} F &= \left< \frac{\partial S}{\partial u}, \frac{\partial S}{\partial v} \right> \\ \frac{\partial F}{\partial v} &= \left<\frac{\partial^2 S}{\partial u \partial v} , \frac{\partial S}{\partial v} \right> + \left<\frac{\partial S}{\partial u} , \frac{\partial^2 S}{\partial v^2 } \right> \\ \frac{\partial F}{\partial v} &= \left<\frac{\partial S}{\partial u} , \frac{\partial^2 S}{\partial v^2 } \right> + \left<\frac{\partial^2 S}{\partial v \partial u} , \frac{\partial S}{\partial v} \right> \\ \frac{\partial F}{\partial v} &= \left<\frac{\partial S}{\partial u} , \Gamma_{22}^{1} \frac{\partial S}{\partial u} + \Gamma_{22}^{2} \frac{\partial S}{\partial v} + Q^{4}(u,v)  \hat{n} \right> + \frac{1}{2}\frac{\partial G}{\partial u} \\ \frac{\partial F}{\partial v} - \frac{1}{2}\frac{\partial G}{\partial u} &= E\Gamma_{22}^{1} + F\Gamma_{22}^{2} \end{align*}} $$

Como S(u,v) es una superficie continua entonces las segundas derivadas parciales son las mismas. Por consiguiente los siguientes símbolos de Christoffel son simétricos en la parte del subíndice.

$$ {\large \begin{equation*} \Gamma_{12}^{1} = \Gamma_{21}^{1} \hspace{1cm},\hspace{1cm} \Gamma_{12}^{2} = \Gamma_{21}^{2} \end{equation*}} $$

Entonces podemos agrupar los símbolos de Christoffel de la siguiente forma: 

$$ {\large \begin{align*} E\Gamma_{11}^{1} + F\Gamma_{11}^{2} &= \frac{1}{2}\frac{\partial E}{\partial u} \\ F\Gamma_{11}^{1} + G\Gamma_{11}^{2} &= \frac{\partial F}{\partial u} - \frac{1}{2}\frac{\partial E}{\partial v} \\ \\ E\Gamma_{12}^{1} + F\Gamma_{12}^{2} &= \frac{1}{2}\frac{\partial E}{\partial v} \\ F\Gamma_{21}^{1} + G\Gamma_{21}^{2} &= \frac{1}{2}\frac{\partial G}{\partial u} \\ \\ F\Gamma_{22}^{1} + G\Gamma_{22}^{2} &= \frac{1}{2}\frac{\partial G}{\partial v} \\ E\Gamma_{22}^{1} + F\Gamma_{22}^{2} &= \frac{\partial F}{\partial v} - \frac{1}{2}\frac{\partial G}{\partial u} \end{align*}} $$

Los símbolos de Christoffel pueden despejarse haciendo uso del álgebra lineal, ya sea usando la matriz inversa o la regla de Cramer respectivamente.

$$ {\large \begin{equation*} \begin{bmatrix} \Gamma_{11}^{1} & \Gamma_{11}^{2} \\ \\ \Gamma_{12}^{1} & \Gamma_{12}^{2} \\ \\ \Gamma_{22}^{1} & \Gamma_{22}^{2}  \end{bmatrix} = \frac{1}{2(EG - F^2)} \begin{bmatrix} G\frac{\partial E}{\partial u} - 2F\frac{\partial F}{\partial u} + F\frac{\partial E}{\partial v} & 2E\frac{\partial F}{\partial u} - E\frac{\partial E}{\partial v} - F\frac{\partial E}{\partial u} \\ \\ G\frac{\partial E}{\partial v} - F\frac{\partial G}{\partial u} & E\frac{\partial G}{\partial u} - F\frac{\partial E}{\partial v} \\ \\ 2G\frac{\partial F}{\partial v} - G\frac{\partial G}{\partial u} - F\frac{\partial G}{\partial v} & E\frac{\partial G}{\partial v} - 2F\frac{\partial F}{\partial v} + F\frac{\partial G}{\partial u} \end{bmatrix} \end{equation*}} $$

Donde los valores de E, F y G son los coeficientes de la primera forma fundamental de la superficie S(u,v) respectivamente.

Aplicación de los símbolos de Christoffel

Si tenemos una superficie de revolución $S(u,v) = (f(u)\cos(v), f(u)\sin(v), h(u))$ donde los coeficientes de la primera forma fundamental de S(u,v) son:

$${\large \begin{align*} E &= \left( \frac{df}{du} \right)^2 + \left( \frac{dh}{du} \right)^2 \\ F &= 0 \\ G &= f^2(u) \end{align*} } $$

Las derivadas parciales de los coeficientes de la primera forma fundamental son:

$${\large \begin{align*} \frac{\partial E}{\partial v} &= 0 & \frac{\partial E}{\partial u} &= 2\left( \frac{df}{du}\frac{d^2f}{du^2} + \frac{dh}{du}\frac{d^2h}{du^2}\right) \\   \frac{\partial F}{\partial v} &= 0 & \frac{\partial F}{\partial u} &= 0 \\ \frac{\partial G}{\partial v} &= 0 & \frac{\partial G}{\partial u} &= 2f(u)\frac{df}{du}\end{align*} }$$

Por lo tanto los símbolos de Christoffel de una superficie de revolución es igual a:

$${\large \begin{equation*} \begin{bmatrix} \Gamma_{11}^{1} & \Gamma_{11}^{2} \\ \\ \Gamma_{12}^{1} & \Gamma_{12}^{2} \\ \\ \Gamma_{22}^{1} & \Gamma_{22}^{2}  \end{bmatrix} =  \begin{bmatrix} \frac{f(u)\left( \frac{df}{du}\frac{d^2f}{du^2} +  \frac{dh}{du}\frac{d^2h}{du^2}\right)}{\left( \frac{df}{du} \right)^2 + \left( \frac{dh}{du} \right)^2 }  &  0 \\ \\ 0  & \frac{d f(u)}{\partial u} \\ \\ -\frac{f(u) \frac{d f(u)}{\partial u}}{ \left( \frac{df}{du} \right)^2 + \left( \frac{dh}{du} \right)^2 }  & 0 \end{bmatrix} \end{equation*}} $$