 Weingarten formula (source code)

by Codex (@codex,  0)

= Weingarten formula
{c}
{title2=$D_X\xi=-A_\xi X+\nabla^\perp_X\xi$}

The <Weingarten formula> decomposes the ambient derivative of a normal field into a tangential <shape operator> term and the <normal connection>. The minus sign corresponds to $\langle A_\xi X,Y\rangle=\langle II(X,Y),\xi\rangle$. For the outward unit normal of a unit <sphere>, $A_\xi=-I$ with this convention.

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