This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal time-like unit vector $\boldsymbol{n}$. Following Corichi's notes, he defines the extrinsic curvature as \begin{align} K_{ab}=\boldsymbol{n}\cdot ^{(4)}\nabla_a e_b\,, \end{align} where $\cdot$ is the dot product. In Eq. (3.27) of his notes, he writes the three-dimensional Christoffel symbol can be written as \begin{align} ^{(3)}\Gamma_{abc}=e_a\cdot \,^{(3)}\nabla_c e_b\,. \end{align} With these two equations, he says that it is possible to write the covariant derivative of the tetrad $^{(4)}\nabla_a e_b$, in terms of the induced components of the hypersurface $e_a$ and the normal vector $\boldsymbol{n}$ in form \begin{align} ^{(4)}\nabla_a e_b=-K_{ab}\boldsymbol{n}+^{(3)}\Gamma^{c}_{ba}e_c\,, \end{align} which is known as Gauss equation, obtained in Eq. (3.32). Although I project the normal vector in the extrinsic curvature equation, I obtain $K_{ab}\boldsymbol{n}=\underbrace{\boldsymbol{n}\cdot\boldsymbol{n}}_{-1}\cdot\,^{(4)}\nabla_a e_b=-^{(4)}\nabla_a e_b$, and no Christoffel symbol contribution. How this term appear?
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