CR Submanifolds of Maximal CR Dimension in a Complex Hopf Manifold |
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Authors: | Elisabetta Barletta |
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Institution: | (1) Dipartimento di Matematica, Università degli Studi della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italia |
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Abstract: | We study CR submanifolds M in a Hopf manifold (C
H
N
( ), J
0, g
0) with the Boothby metric g
0,of maximal CR dimension. Any such M is a CR manifold ofhypersurface type, although embedded in higher codimension, and itsanti-invariant distribution H(M) is spanned by a unit vectorfield U. We classify the CR submanifolds M for which = –J
0
Uis parallel in the normal bundle under assumptions on thespectrum of the Weingarten operator a
. We show that (1) ifa
(U) = (1/2)A (where A is the anti-Lee vector) andM fibres in tori over a CR submanifold of the complex projectivespace, then M lies on the (total space of the) pullback of the Hopf fibration via S C
P
N – 1, for some geodesic hypersphere S, and (2) if a
(U)= 0 and Spec(a
) = {0, c}, for some c R {0}, then M is locally a Riemannian product of totally geodesicsubmanifolds. |
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Keywords: | complex Hopf manifold Boothby metric CR submanifold Weingarten operator geodesic hypersphere |
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