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?
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1$\begingroup$ Could you give some more definitions or a reference where the definitions are given? Are you looking for a CR hypersurface? What do you mean by 'a hypersurface of the shape operator'? Your question looks as though you have left out some words here and there, and I can't figure out what you are asking. $\endgroup$– Robert BryantCommented Nov 28, 2011 at 12:52
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$\begingroup$ We say that the n-dimensional submanifold is a CR submanifold of maximal CR dimension if the complex dimension of the holomorphic tangent space $H_{x}=T_{x}\intersection JT_{x}$, $T_{x}$ is the tangent space, equals $\frac{n-1}{2}$.Let's say that the ambient manifold is a complex space form of dimension $n+p$.So, there is only one normal vector which J maps to the tangent vector. J is an almost complex structure. And I'm observing some conditions on the shape operator $A$ of that special normal vector.I need one example with $A=0$.A hypersurface is a CR submanifold of maximal CR dimension. $\endgroup$– MirjanaCommented Nov 29, 2011 at 14:26
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$\begingroup$ And the definitions are given in a book Djoric, Okumura, CR submanifolds of complex projective space. $\endgroup$– MirjanaCommented Nov 29, 2011 at 14:39
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$\begingroup$ @Mirjana: Thanks, I think I understand now. You didn't mention before that you are considering submanifolds in a complex space form, which is an essential part of the problem, since, for CR submanifolds of general complex manifolds, you don't have a distinguished normal. I still don't know what you mean by "hypersurface of the shape operator equals zero". Do you mean "hypersurface for which the shape operator equals zero"? $\endgroup$– Robert BryantCommented Nov 30, 2011 at 13:07
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1$\begingroup$ Yes, that is, a hypersurface (in complex space form) for which the shape operator equals zero, I don't know how will I find an example, for example in complex space C^{n}. And I would like to have some example to see, because I proved some nonexistence results in non flat complex space forms. And if the shape operator equals zero then all of my conditions are satisfied. $\endgroup$– MirjanaCommented Dec 1, 2011 at 0:28
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2 Answers
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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.
answered Dec 6, 2011 at 13:36
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$\begingroup$ So we can not find these examples in nonflat complex space forms. Thank you! $\endgroup$– MirjanaCommented Dec 7, 2011 at 5:16
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$\begingroup$ @Mirjana: I'm not so sure. There are certainly many curves $\gamma:(a,b)\to\mathbb{CP}^n$ that satisfy $\gamma'(t)\cdot\gamma''(t) = J\gamma'(t)\cdot\gamma''(t)=0$ (where $\gamma''(t)$ means just the covariant derivative of $\gamma'(t)$ along the curve). For example, any constant speed curve whose image lies in $\mathbb{RP}^n$ will satisfy this. Won't these be examples? As for higher dimensional examples, I wouldn't guess without checking the structure equations. $\endgroup$ Commented Dec 7, 2011 at 13:06
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If I'm not misunderstanding your question, this might be the example you're looking for: the standard CR-product of $\Bbb CP^n$ and $\Bbb RP^1$ immersed in $\Bbb CP^{2n+1}$.
