Let be $M^n$ an $n$-dimensional smooth manifold and let $$X(\cdot, t): M^n \longrightarrow \mathbb{R}^{n+1}$$
be a one-parameter family of smooth hypersurface immersions in $\mathbb{R}^{n+1}$. Xi-Phing Zhu states in his book on page 16 that $$\frac{\partial^2 X}{\partial x^i \partial x^j} = \Gamma^k_{ij} \frac{\partial X}{\partial x^k} + h_{ij} \overrightarrow{n},$$ where $\overrightarrow{n}$ is an unit normal on $X(\cdot, t)$ and $h_{ij}(x,t) = \langle \overrightarrow{n}, \frac{\partial^2 X(x,t)}{\partial x^i \partial x^j} \rangle$, is valid by the Weingarten's equation, but I didn't understand how exactly the Weingarten's equation helps me to show that. Can anyone explain why this is valid?
Thanks in advance!
P.S.: The author assumes the Einstein summation convention in the book.
The tangential component of this vector equation is precisely the definition of the Christoffel symbols $\Gamma^k_{ij}$, just as the normal component is the definition of the second fundamental form of the hypersurface.