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Riemannian Geometry of Contact and Symplectic Manifolds

08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

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Journal Article
TL;DR: In this paper, it was shown that the CR-Yamabe equation on the Heisenberg group has infinitely many changing-sign solutions, and the result was generalized to any compact contact manifold of K-contact type.
Abstract: In this paper we prove that the CR-Yamabe equation on the Heisenberg group has innitely many changing-sign solutions. By means of the Cayley transform we will set the problem on the sphere S 2n+1 ; since the functional I associated with the equation does not satisfy the Palais-Smale compactness condition, we will nd a suitable closed subspace X on which we can apply the minmax argument for IjX. We generalize the result to any compact contact manifold of K-contact type.

21 citations


Cites background from "Riemannian Geometry of Contact and ..."

  • ...Moreover on S2n+1 we have the following crucial property of commutation T ∆θ0 = ∆θ0 T (5) As we will see in the next section the sphere (S2n+1, θ0) is a particular case of K-contact manifold according the definition in [4]....

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  • ...Here we give the basic definitions and properties, for a comprehensive presentation concerning the subject we refer the reader to [4]....

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01 Jan 2010
TL;DR: In this article, an example of a three dimensional flat paracontact metric manifold with respect to the Levi-Civita connection is constructed, and it is shown that no such manifold exists for odd dimensions greater than or equal to five.
Abstract: An example of a three dimensional flat paracontact metric manifold with respect to Levi-Civita connection is constructed. It is shown that no such manifold exists for odd dimensions greater than or equal to five.

21 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied affine biharmonic curves in model spaces of Thurston geometry except Sol and showed that every 3-dimensional Riemannian manifold with 4-dimensional isometry group admits a normal almost contact structure compatible to the metric.
Abstract: Every 3-dimensional Riemannian manifold with 4-dimensional isometry group admits a normal almost contact structure compatible to the metric. In this paper we study affine biharmonic curves in model spaces of Thurston geometry except Sol.

21 citations

Journal ArticleDOI
TL;DR: In this article, a Riemanian metric on the tangent bundle T(M) of a manifold M which generalizes Sasaki metric and Cheeger Gromoll metric is studied.
Abstract: In this paper we study a Riemanian metric on the tangent bundle T(M) of a Rieman- nian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T(M) a structure of locally conformal almost Kahlerian manifold. This is the natural gener- alization of the well known almost Kahlerian structure on T(M). We found conditions under which T(M) is almost Kahlerian, locally conformal Kahlerian or Kahlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki met- ric from T(M). Moreover, we found that this map preserves also the natural almost contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively.

21 citations

Journal ArticleDOI
Toru Kajigaya1
TL;DR: In this article, the authors investigated compact Legendrian submanifolds in Sasakian manifolds, which have extremal volume under Legendrian deformations, and derived the second variational formula for the volume of $L$ under Legendian deformations.
Abstract: In this paper, we investigate compact Legendrian submanifolds $L$ in Sasakian manifolds $M$, which have extremal volume under Legendrian deformations. We call such a submanifold $L$-minimal Legendrian submanifold. We derive the second variational formula for the volume of $L$ under Legendrian deformations in $M$. Applying this formula, we investigate the stability of $L$-minimal Legendrian curves in Sasakian space forms, and show the $L$-instability of $L$-minimal Legendrian submanifolds in $S^{2n+1}(1)$. Moreover, we give a construction of $L$-minimal Legendrian submanifolds in ${\boldsymbol R}^{2n+1}(-3)$.

21 citations