Complex contact manifolds and circle actions

2015

In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold. Also, we give some new equailities for Ricci curvature of normal complex contact metric manifolds.

Journal of Geometry, 2008

Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the nongradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HP CV ). Finally we show that, if a (k, μ)-contact manifold admits a non-zero HP CV , then it is either Sasakian or locally isometric to E 3 or E n+1 × S n (4).

2006

We study the differential geometric properties of n-dimensional real submanifolds M of complex space forms M , whose maximal holomorphic tangent subspace is (n− 1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space. Using certain condition on the induced almost contact structure F and on the second fundamental form h of these submanifolds, which is sufficient for F to be the contact one, we give a classification of such submanifolds M and we obtain new characterizations of some model spaces in complex space forms.

Tohoku Mathematical Journal, 1979

Differential Geometry and its Applications, 2012

Generalized contact pairs were introduced in Poon and Wade (2011) [25]. In this paper, we carry out a detailed study of geometric properties of these structures. First, we give geometric conditions expressing the integrability of a generalized contact pair. Then, we use them to obtain insights into the characteristic foliation of a generalized contact manifold. Finally we show that, locally, any smooth manifold endowed with a generalized contact pair is equivalent to the product of an almost cosymplectic manifold whose associated 2-form is closed by a generalized complex manifold.

Mathematische Nachrichten, 1999

We treat n‐dimensional real submanifolds of complex projective spaces in the case when the maximal holomorphic tangent subspace is (n ‐ 1)‐dimensional. In particular, we study the case when the induced almost contact structure on a submanifold is contact, we establish a few characteristics of the shape operator with respect to the distinguished vector field and we give one characterization of a contact submanifold in this case

ISRN Geometry, 2013

We studyDa-homothetic deformations ofK-contact manifolds. We prove thatDa-homothetically deformedK-contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature ofDa-homothetically deformedK-contact manifolds.

Georgian Mathematical Journal

We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat. We also find some necessary conditions for a K-contact manifold to be ϕ-contact conformally flat. Then we give a structure theorem for ϕ-contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ-contact conformally flat.

arXiv (Cornell University), 2009

Any connected Fano complex contact manifold is contactly biholomorphic to a kählerian C-space of Boothby type, that is, the adjoint variety of a finitely dimensional complex simple Lie algebra of rank greater than or equal to 2, with the canonical complex contact structure.

Mathematics

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of C C -totally real submanifolds of maximum dimension satisfying the equalities.