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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 $\nabla_X Y$?

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1 Answer 1

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 $(\nabla_X Y)_p := \nabla_{Xp}Y$.

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Thanks. So $X \in T_x\Gamma$ I assume. What's $X_p$ mean? Is there any relation to the Levi-Civita connection? Is this connection you defined the natural one? –  hopo2 Sep 21 '12 at 13:25
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@hopo2: part of your follow up question has been asked as a standalone one here. –  Willie Wong Nov 30 '12 at 11:18

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