The surface gradient of a function defined on a surface $\Gamma \subset \mathbb{R}^n$ is defined
$$\nabla_{\Gamma} f = \nabla f – (\nabla f \cdot N)N$$
where $N$ is the unit normal on $\Gamma.$
How do I obtain this from the covariant derivative operator $\nabla_X Y$?
Best Answer
For a tangent vector $\bf v$ at a point $x\in\Gamma$, and a smooth (real valued) function $f$ defined on an open neighborhood of $x$, one can define $\nabla_{\bf v}f:=\langle {\bf v},\nabla_\Gamma f\rangle$ (the scalar product). Note that both $\bf v$ and $\nabla_\Gamma f$ already lie in the tangent space of $\Gamma$, and hence only the scalar product of $T_x\Gamma$ is used.
Then, for a vector field $Y$ (on $\Gamma$), if given by coordinate functions $(f_i)_{i=1}^n$, define $\nabla_{\bf v}Y := (\nabla_{\bf v} f_i)_i$.
Finally, for a point $x\in\Gamma$, set $(\nabla_X Y)_p := \nabla_{Xp}Y$.