I am recently taking the undergraduate version of differential geometry. This week we finished the Compatibility Equations. Actually the contents given in the lecture are quite different from the book I read, Differential Geometry of Curves and Surfaces, by De Carmo.
In this book, the author takes the following equations as the Gauss Formula and Mainardi-Codazzi Equations. $$(\Gamma_{12}^2)_u - (\Gamma_{11}^2)_v + \Gamma_{12}^1\Gamma_{11}^2 + \Gamma_{12}^2\Gamma_{12}^2 -\Gamma_{11}^2\Gamma_{22}^2 - \Gamma_{11}^1\Gamma_{12}^2 = -EK,$$ $$e_v - f_u = e\Gamma_{12}^1 + f(\Gamma_{12}^2 - \Gamma_{11}^1) + g\Gamma_{11}^2,$$ $$f_v - g_u = e\Gamma_{22}^1 + f(\Gamma_{22}^2 - \Gamma_{12}^1) + g\Gamma_{12}^2.$$ where $\Gamma_{ij}^K$ is the Christoffel Symbols and $E,F,G,e,f,g$ are the coefficients of the first fundamental form and the second fundamental form respectively.
The version given in the lecture: $$\langle X_u,RX_v \rangle = K(EG-F^2),$$ $$\nabla_{\partial u}(\zeta X_v) = \nabla_{\partial v} (\zeta X_u),$$ where $K$ is the Guassian curvature, $\nabla V = (dV)^T = dV - \langle dV, N \rangle N$ and $\zeta$ is the weingarten operator. $X$ is the corresponding parametrization compatible with the orientation. Moreover, $R$ is the curvature tensor.
In fact, I do not know what is the meaning of tensor, so let me just give the definition the lecturer presented.
Definition. $$RV = \nabla_{\partial u}\nabla_{\partial v}V - \nabla_{\partial v}\nabla_{\partial u}V.$$
Could anyone explain the connection between the two sets of equations? How to show they are equivalent? Thanks.
