I need an example of a CR submanifold of maximal CR dimension with the shape operator of the distinguished normal equals zero, or a hypersufrace of the shape operator equals zero. Can anyone help me?



shape operator of a hypersurface or a CR submanifolds of maximal CR dimension
Well, there are examples, but it appears that they are all essentially trivial.


Take the simplest case of a curve in $\mathbb{C}^n$, which automatically has maximal CR dimension. (Of course, that CR dimension is $0$). Let's say that it is parametrized at constant speed, $\gamma:(a,b)\to \mathbb{C}^n$. Then the condition you require is that $J\gamma'(t)\cdot\gamma''(t) = 0$, along with $\gamma'(t)\cdot\gamma''(t)=0$. This is two second-order equations for $\gamma$, and it follows that the general solution depends on $2(n{-}1)$ arbitrary functions of one variable.


For higher dimensional examples, you can do the following: Write $\mathbb{C}^m$ as an orthogonal direct sum $\mathbb{C}^{p}\oplus\mathbb{C}^{q}$; let $\gamma\subset\mathbb{C}^{p}$ be a curve of the above type that is nondegenerate, i.e., its first $2p$ derivatives are linearly independent everywhere (which is the generic case for these curves); and, finally, let $S\subset\mathbb{C}^{q}$ be a (connected) complex submanifold of complex dimension $n$ that does not lie in any proper affine subspace of $\mathbb{C}^{q}$. Then the product $\gamma\times S\subset \mathbb{C}^m$ is of dimension $2n{+}1$, is maximally complex, and satisfies your conditions. It also does not lie in any proper complex affine subspace of $\mathbb{C}^m$.


A little work with the structure equations of $\mathbb{C}^m$ shows that the `generic' CR manifold of dimension $2n{+}1$ and complex dimension $n$ in $\mathbb{C}^m$ that satisfies your condition on the second fundamental form in the direction of the distinguished normal is one of these product examples. In particular, there is no example satisfying your criteria that is CR-nondegenerate.


