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I saw in a talk that a surface gradient of $f:M \to \mathbb{R}$ where $M$ is a hypersurface in $\mathbb{R}^n$ defined as $$\nabla_M f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal vector on $M$ and $\nabla$ is the ordinary gradient.

I just started learning about the connection/covariant derivative on a manifold and am wondering about the link. Is the surface gradient as defined above just a choice of a particular connection? Does it have anything to do with the Levi-Civita connection?

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    This question has been asked before it seems: http://math.stackexchange.com/q/200223/266952012-11-26
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    Should this really be closed? The question @begeistzwerst links to does not contain the answers to everything AC21 has asked about.2012-11-26
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    Actually I did not intend to close it. I only wanted to draw attention to the other question. I am not sure what's best practice in this situation.2012-11-26

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