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Register a new userLegendrian cone structures and contact prolongations
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Authors
Jun-Muk Hwang
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We study a cone structure ${mathcal C} subset {mathbb P} D$ on a holomorphic contact manifold $(M, D subset T_M)$ such that each fiber ${mathcal C}_x subset {mathbb P} D_x$ is isomorphic to a Legendrian submanifold of fixed isomorphism type. By characterizing subadjoint varieties among Legendrian submanifolds in terms of contact prolongations, we prove that the canonical distribution on the associated contact G-structure admits a holomorphic horizontal splitting.
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Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollar). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale-Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale-Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic Kahler orbifolds to the algebro-geometric conjecture that claims that for an algebraic surface with $b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves of genus g>0 spanning the (rational) homology.
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${mathcal O}_{{mathbb P}^1}(1)^{oplus p} oplus {mathcal O}_{{mathbb P}^1}^{oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x in C,$ which is the germ of submanifolds ${mathcal C}^C_x subset {mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$. When $D subset TX$ is a contact distribution, a well-known necessary condition is that ${mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D subset TX$ and an unbendable rational curve $C subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$ at some point $xin C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.
We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.
We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Mreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.
An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations reduce to those for the characteristic field to be a harmonic section of the unit tangent bundle. These include trans-Sasakian structures, and certain nearly cosymplectic structures. On the other hand, we obtain examples where the characteristic field is harmonic but the almost contact structure is not. Many of our examples are obtained by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures.
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