Some remarks on R-contact flows

Rukimbira, Philippe

doi:10.1007/bf00773454

Cited by 21 publications

(12 citation statements)

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“…Note that the closed characteristics of M correspond to periodic integral curves of JN restricted to M, where J is the standard symplectic matrix and N the outward normal to M. This conjecture was proved by Viterbo [21] when the ambient symplectic manifold is the Euclidean space with its canonical symplectic form. The conjecture is also true if the contact hypersurface is Sasakian, as shown by Rukimbira [16]. Recently, Ghosh, Koufogiorgos and Sharma [12] proved that the mean curvature of the contact hypersurface M of a Kaehler Einstein manifold is constant iff the characteristic vector field of M is an eigenvector of the Ricci tensor.…”

Section: Introductionmentioning

confidence: 81%

Contact hypersurfaces of Kaehler manifolds

Sharma

2003

Journal of Geometry

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Section: Introductionmentioning

confidence: 81%

Contact hypersurfaces of Kaehler manifolds

Sharma

2003

Journal of Geometry

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“…The following result was first proved by Rukimbira [101] in a slightly different, although equivalent, setting and later by Itoh [63] in the K-contact setting. Proof.…”

Section: K-contact Structuresmentioning

confidence: 75%

The geometry and topology of 3-Sasakian manifolds.

1994

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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This thesis is concerned with the topology and geometry of Sasakian manifolds. Sasaki structures consist of certain contact forms equipped with special Riemannian metrics. Sasakian manifolds relate to arbitrary contact manifolds as Kählerian or projective complex manifolds relate to arbitrary symplectic manifolds. Therefore, Sasakian manifolds are the odd-dimensional analogs of Kähler manifolds. In the first part of the thesis we discuss some geometric invariants of Sasaki structures. Specifically, the socalled basic Hodge numbers, the type and their relation to the underlying contact and almost contact structures are discussed. We produce many pairs of negative Sasakian structures with distinct basic Hodge numbers on the same differentiable manifold in any odd dimension larger than 3. In the second part of the thesis we discuss topological properties of Sasakian manifolds, focussing particularly on the fundamental groups of compact Sasakian manifolds. In parallel with the theory of Kähler and projective groups, we call these groups Sasaki groups. We prove that any projective group is realizable as the fundamental group of a compact Sasakian manifold in every odd dimension larger than three. Similarly, every finitely presentable group is realizable as the fundamental group of a compact K-contact manifold in every odd dimension larger than three. Nevertheless, Sasaki groups satisfy some very strong constraints, some of which are reminiscent of known constraints on Kähler groups. We show that the class of Sasaki groups is not closed under direct products and that there exist Sasaki groups that cannot be realized in arbitrarily large dimension. We prove that Sasaki groups behave similarly to Kähler groups regarding their relation to 3-manifold groups and to free products.

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“…Suppose M has a K-contact form which is not almost regular. By the structure Theorem 2.1 in [RU3] and Proposition 2 of this paper, M would carry a finite number of closed characteristics. By Proposition 1, there are exactly 2 closed characteristics which are critical circles of a clean function, one of index 0 and another one of index 2.…”

Section: Application To Brieskorn Manifoldsmentioning

confidence: 95%