Let $M\subseteq \mathbb{R}^3$ be a regular surface and $p\in M$. The Riemann curvature tensor is defined by: $$\begin{array}{rcll} R_p:&T_pM\times T_pM\times T_pM&\longrightarrow &T_pM\\ &(\vec x_i,\vec x_j,\vec x_k)&\longmapsto &R_p(\vec x_i,\vec x_j)\vec x_k= \sum\limits_{l=1}^2 R_{ijk}^l \vec x_l \end{array}$$ with: $$R_{ijk}^l =\frac{\partial \Gamma^l_{jk}}{\partial u^i} -\frac{\partial \Gamma_{ik}^l}{\partial u^j}+\sum_{h=1}^n\Big(\Gamma_{jk}^h\Gamma_{ih}^l-\Gamma_{ij}^h\Gamma_{jh}^l\Big)$$ Is this definition equivalent to the first or second fundamental form for $M$? If not, what is the geometric interpretation?
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