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Consider the sphere $S^n$. By using the stereographic projection we can identify $S^n \setminus N$ with $\mathbb{R}^n$, where $N$ is the North pole of $S^n$. The metric then is given by $\frac{dx^2}{(1+x^2)^2}$. Now we consider the graph of a smooth function $f$ on an open subset of $\mathbb{R}^{n-1}$ as a subset of $\mathbb{R}^n$ with the above metric. Can you tell me what the mean curvature of this graph is in terms of $f$, or can you at least name me a book, where I can look this up? Every help would be appreciated.

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    The shape operator is defined using the connection on the ambient manifold, which in this case is just the ordinary directional derivative. The tangent planes (and thus the normal vector) of the graph of $f$ are independent of any metric on the graph of $f$. Thus it not clear (to me) how the shape operator ($X \to D_X n$) changes when the metric on the graph of $f$ changes.2012-12-21

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