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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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06 Mar 2020
TL;DR: In this paper, the authors studied the geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms, and derived vanishing theorems about the null space of the Hodge type Laplacian.
Abstract: We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenbock type curvature operator on tensors, prove the Weitzenbock type decomposition formula, and derive the Bochner–Weitzenbock type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.
2 citations
TL;DR: In this paper, the authors introduced the notions of semi-slant submanifolds of an almost contact 3-structure manifold and gave some examples and characterization theorems about them.
Abstract: In this paper we introduce the notions of semi-slant and bi-slant submanifolds of an almost contact 3-structure manifold. We give some examples and characterization theorems about these submanifolds. Moreover, the distributions of semi-slant submanifolds of 3-cosymplectic and 3-Sasakian manifolds are studied. ϕi)i=1;2;3 and the vector elds should be slant or invariant with respect to all of the ϕi's. Therefore, it is a generalization of invariant, anti-invariant, slant, semi-slant, and bi-slant submanifolds in almost contact metric 3-structures and we denote them by 3- semi-slant and 3-bi-slant submanifolds. Following the approaches of (3, 13), we characterized 3-bi-slant and 3-semi-slant submanifolds and studied geometric properties of distributions of these submanifolds where the ambient manifolds are 3-Sasakian or 3-cosymplectic. It should be noted that, in the denition of semi-slant
2 citations
TL;DR: In this article , it was shown that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field, then the manifold is flat.
Abstract: The aim of the present paper is to characterize almost co-K?hler manifolds
whose metrics are the Riemann solitons. At first we provide a necessary and
sufficient condition for the metric of a 3-dimensional manifold to be
Riemann soliton. Next it is proved that if the metric of an almost
co-K?hler manifold is a Riemann soliton with the soliton vector field ?,
then the manifold is flat. It is also shown that if the metric of a (?,
?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the
soliton is expanding and ?, ?, ? satisfies a relation. We also prove that
there does not exist gradient almost Riemann solitons on (?, ?)-almost
co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton
on a three dimensional almost co-K?hler manifold is ensured by a proper
example.
2 citations
TL;DR: In this article, the authors proved an optimal inequality for the contact CR $$\delta $$ウス -invariant on contact CR-submanifolds in Sasakian space forms.
Abstract: The theory of $$\delta $$
-invariants was initiated by Chen (Arch Math 60:568–578, 1993) in order to find new necessary conditions for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Chen (Int J Math 23(3):1250045, 2012) defined a CR $$\delta $$
-invariant $$\delta (D)$$
for anti-holomorphic submanifolds in complex space forms. Afterwards, Al-Solamy et al. (Taiwan J Math 18:199–217, 2014) established an optimal inequality for this invariant for anti-holomorphic submanifolds in complex space forms. In this article, we prove an optimal inequality for the contact CR $$\delta $$
-invariant on contact CR-submanifolds in Sasakian space forms. An example for the equality case is given.
2 citations
18 Jul 2014
TL;DR: In this paper, the authors studied homogeneous Riemannian structure tensors under the framework of reduction under a group H of isometries, which is a normal subgroup of the symmetries associated with the reducing tensor.
Abstract: The goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kahler.
2 citations