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Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer normal $N$). Let $P:T(S)\rightarrow T(S)$ be the operator defined by $P(X)=J(X)-\langle J(X),N\rangle N$. I am interested in any reasonable geometric interpretation of the operator $F_t:=A+tP$ for $t\in\mathbb{R}$ and the zeros of $\det F_t$. It looks like a kind of complex/contact deformation of the shape operator. But what essential geometric objects does it measure?

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You need to define $\alpha$; I think I can guess, but it would be better to know. –  Robert Bryant Mar 17 '13 at 13:45
The 1-form $\alpha$ is given by the standard scalar product $\langle J(X), N\rangle$ –  Serj Mar 17 '13 at 17:07

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