Contact hypersurfaces of Kaehler manifolds

Abstract

Contact hypersurfaces of a Kaehler manifold have been characterized and classified, assuming the second fundamental form to be Codazzi (in particular, parallel). We have also discussed the special cases when the ambient space is a (i) Calabi-Yau manifold and (ii) a complex space-form. (2000): 53C15, 53C55, 53C42. Mathematics Subject Classification

Contact Hypersurfaces of a Bochner–Kaehler Manifold

“…Using this in (16) we find ϕQξ = ϕQξ. Finally, using this in (15) we conclude that ϕQξ = 0 which implies (b), and complete the proof. Proof Replacing Y, Z by ϕY, ϕZ respectively, in (14) and then using (10) we get (a).…”

“…Okumura [13] studied and classified such hypersurfaces, mainly when the ambient space is a complex space-form. Generalizing the following result of Sharma [15] "The contact metric hypersurface of a complex space-form is a (k, µ)-contact manifold", we prove the following main result of this paper.…”

“…The equations (15) and (16) show that (a) is equivalent to (b). Contracting the Codazzi equation: Then, we go back to equations (17) and (18) and use the formula (divhϕ)Y = S(Y, ξ) − 2nη(Y ) once again, gettingS(X, N ) = S(ϕX, ξ) = −g(ϕQξ, X).…”

Almost Contact Metric Structures on the Hypersurface of Almost Hermitian Manifolds

“…In [171], Sharma characterizes and classifies contact hypersurfaces of a Kählerian manifold, whose second quadratic form is a Codazzi form (and, in particular, is parallel). The case where the ambient Kählerian manifold is a complex spatial form or a Calabi-Yau manifold is studied in detail.…”

On Semiparallel and Weyl-semiparallel Hypersurfaces of Kaehler Manifolds

“…From (9) and (10), (η, ξ, ϕ, g) defines an almost contact metric structure on M . Differentiating (10) along M , using (11) and (12), and comparing tangential parts we get…”