Papers by Philippe Rukimbira
arXiv (Cornell University), 2023
Generalized $(\kappa ,\mu )$ structures occur in dimension 3 only. In this dimension 3, only K-co... more Generalized $(\kappa ,\mu )$ structures occur in dimension 3 only. In this dimension 3, only K-contact structures can occur as generalized Eta-Einstein. On closed manifolds, Eta-Einstein, K-contact structures which are not D-homothetic to Einstein structures are almost regular. We also construct examples of compact, generalized Jacobi $(\kappa ,\mu )$-structures.
arXiv (Cornell University), Jul 13, 2022
Contact metric (κ, µ)-spaces are generalizations of Sasakian spaces. We introduce a weak (κ, µ) c... more Contact metric (κ, µ)-spaces are generalizations of Sasakian spaces. We introduce a weak (κ, µ) condition as a generalization of the K-contact one and show that many of the known results from generalized Sasakian geometry hold in the weaker generalized K-contact geometry setting. In particular, we prove existence of K-contact and (κ, µ = 2)-structures under some conditions on the Boeckx invariant. MSC: 57C15, 53C57 1 Basic properties of contact metric structures A contact form on a 2n + 1-dimensional manifold M is a one-form η such that η ∧ (dα) n is a volume form on M. Given a contact manifold (M, η), there exist tensor fields (ξ, φ, g), where g is a Riemannian metric and ξ is a unit vector field, called the Reeb field of η and φ is an endomorphism of the
African Diaspora Journal of Mathematics. New Series, Sep 20, 2009
This paper contains a characterization of Reeb vector fields of K-contact forms in terms of J-hol... more This paper contains a characterization of Reeb vector fields of K-contact forms in terms of J-holomorphic embeddings into the tangent unit sphere bundle. A consequence of this characterization is that these vector fields are critical points of a volume and an energy functionals defined on the set of unit vector fields. Reeb vector fields on closed, K-contact Einstein manifolds are absolute minimizers for the energy functional with a mean curvature correction. On odd-dimensional Einstein manifolds of positive sectional curvature, these unit vector fields are characterized by their minimizing property. It is also proved that any closed flat contact manifold admits a parallelization by three critical unit vector fields, one parallel (hence minimizing), the other two are Reeb vector fields of contact forms, not Killing and not minimizers of any of the volume or the energy functionals.
Journal of Geometry, Jul 1, 1994
ABSTRACT We prove that on a compact manifold, a contact foliation obtained by a smallC 1 perturba... more ABSTRACT We prove that on a compact manifold, a contact foliation obtained by a smallC 1 perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC 1-close to almost regular contact forms.
Proceedings of the American Mathematical Society, Dec 1, 1995
We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1)-dimensional manifol... more We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1)-dimensional manifold admits two closed cha^cteristics. This solves a particular case of a generalized Weinstein conjecture. Recently, Hofer [HOF] proved the Weinstein conjecture for S3 and for any compact 3-manifold M with n2(M) /0 or the given contact form is overtwisted (see Eliashberg [ELI] for the definition). Observe that the closed characteristic Hofer finds must be a contractible loop.
Journal of Geometry, 1994
ABSTRACT We prove that on a compact manifold, a contact foliation obtained by a smallC 1 perturba... more ABSTRACT We prove that on a compact manifold, a contact foliation obtained by a smallC 1 perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC 1-close to almost regular contact forms.
Proceedings of the American Mathematical Society, 1995
We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1)-dimensional manifol... more We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1)-dimensional manifold admits two closed cha^cteristics. This solves a particular case of a generalized Weinstein conjecture. Recently, Hofer [HOF] proved the Weinstein conjecture for S3 and for any compact 3-manifold M with n2(M) /0 or the given contact form is overtwisted (see Eliashberg [ELI] for the definition). Observe that the closed characteristic Hofer finds must be a contractible loop.
We prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakian manifold M of ... more We prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakian manifold M of rank l is at least equal to 2l−1 provided thatM has an isolated closed characteristic. The result is then used to provide some examples ofK-contact manifolds which are not Sasakian. On a closed, 2n+ 1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifoldM is isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields. 2000 Mathematics Subject Classification: 53D35. 1. Introduction. Contact
We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it... more We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms. 1.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric ... more We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms... more If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact
Contemporary Mathematics, 2003
Journal of Geometry, 2015
Riemannian Topology and Geometric Structures on Manifolds, 2009
ABSTRACT On a closed, (2n + 1)-dimensional Sasakian manifold, we show that either the dimension o... more ABSTRACT On a closed, (2n + 1)-dimensional Sasakian manifold, we show that either the dimension of the 1-nullity distribution N(1) is less than or equal to n, or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold M is isometric to a quotient of the Euclidean sphere under a finite group of isometries.
Journal of Geometry, Jun 30, 1995
ABSTRACT We prove that a contact form with riemannian characteristic flow is K-contact. We also p... more ABSTRACT We prove that a contact form with riemannian characteristic flow is K-contact. We also present a purely riemannian hypothesis which implies the existence of a K-contact form with a prescribed unit Killing vector field as characteristic vector field. Our hypothesis is weaker than that previously presented by Hatakeyama, Ogawa and Tanno.
Proceedings of the American Mathematical Society, 1999
We prove that closed simply connected K-contact manifolds with minimal number of closed character... more We prove that closed simply connected K-contact manifolds with minimal number of closed characteristics are homeomorphic to odd-dimensional spheres.
Bulletin of the Belgian Mathematical Society Simon Stevin, 2000
Proceedings of the American Mathematical Society, 1999
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Papers by Philippe Rukimbira