Certain results on almost contact pseudo-metric manifolds
TL;DR: In this paper, the authors studied the geometry of almost contact pseudo-metric manifold in terms of tensor fields, emphasizing analogies and differences with respect to the contact metric case.
Abstract: We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $$h:=\frac{1}{2}\pounds _\xi \varphi $$
and $$\ell := R(\cdot ,\xi )\xi $$
, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $$\xi $$
-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure $$(\mathcal {H}(M), J, \theta )$$
corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold $$(M, \varphi ,\xi ,\eta ,g)$$
is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure $$(\mathcal {H}(M), J)$$
is integrable and J is parallel along $$\xi $$
with respect to the Bott partial connection.
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TL;DR: In this article, a systematic study of Kenmotsu pseudo-metric manifolds is presented, and the Ricci solitons on these manifolds are considered, and necessary and sufficient conditions for them to have constant curvatures are provided.
Abstract: In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant $\varphi$-sectional curvature, and prove the structure theorem for $\xi$-conformally flat and $\varphi$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $\eta$-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3-manifold admitting a Ricci soliton is of constant curvature $-\varepsilon$.
10 citations
TL;DR: The main purpose of as mentioned in this paper is to study almost cosymplectic pseudo-metric manifold satisfying certain - parallel tensor fields and obtain some results related to the - parallelity of,, and.
Abstract: The main purpose of this paper is to study almost - cosymplectic pseudo metric manifold satisfying certain - parallel tensor fields. We first focus on the concept of almost - cosymplectic pseudo metric manifold and its curvature properties. Then, we obtain some results related to the - parallelity of , , and . Moreover, the deformation of almost - Kenmotsu pseudo metric structure is given. We conclude the paper with an illustrative example of almost - cosymplectic pseudo metric manifold.
2 citations
01 Mar 2021
TL;DR: In this paper, the authors studied the class of almost contact pseudo-metric 3-manifolds of dimension three and derived necessary and sufficient conditions for them to be normal.
Abstract: The purpose of this paper is to study the almost contact pseudo-metric manifolds of dimension three which are normal. We derive certain necessary and sufficient conditions for an almost contact pseudo-metric manifold to be normal. We prove that in a normal almost contact pseudo-metric 3-manifold M of constant curvature k the function $$\beta $$
is harmonic, and if $$\beta =0$$
on M, then the function $$\alpha $$
is harmonic. Furthermore we give the necessary and sufficient condition for normal almost contact pseudo-metric 3-manifold to be $$\beta $$
-Sasakian pseudo-metric manifold. Finally, we study the class of symmetric parallel (0, 2)-tensor field and its consequences.
TL;DR: In this article , the concept of almost α-Kenmotsu pseudo-Riemannian structure and its basic properties were studied and some fundamental formulas and some classification results on such manifolds with CR-integrable structure were presented.
Abstract: This article aims to study almost α-Kenmotsu pseudo-Riemannian structure. We first focus on the concept of almost α-Kenmotsu pseudo-Riemannian structure and its basic properties. Then, we shall prove some fundamental formulas and some classification results on such manifolds with CR-integrable structure. Finally, some illustrative examples of almost α-Kenmotsu pseudo-Riemannian manifold are given.
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Book•
11 Jul 2011
TL;DR: In this article, the authors introduce Semi-Riemannian and Lorenz geometries for manifold theory, including Lie groups and Covering Manifolds, as well as the Calculus of Variations.
Abstract: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.
3,593 citations
Book•
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
1,822 citations
TL;DR: In this paper, the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds was defined, and the torsion and generalized Tanaka-Webster scalar curvature were defined properly.
Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.
291 citations
Book•
30 Aug 2008
TL;DR: The Fefferman Metric and Yamabe Problem were used in this article to define pseudoharmonic maps on CR Manifolds. But they did not consider pseudoeinsteinian manifolds and pseudo-Hermitian immersions.
Abstract: CR Manifolds.- The Fefferman Metric.- The CR Yamabe Problem.- Pseudoharmonic Maps.- Pseudo-Einsteinian Manifolds.- Pseudo-Hermitian Immersions.- Quasiconformal Mappings.- Yang-Mills Fields on CR Manifolds.- Spectral Geometry.
286 citations
TL;DR: In this article, the authors define a Sasakian manifold with pseudo-Riemannian metric, and discuss the classification of the manifold with constant φ-sectional curvatures, and prove that such a manifold is of constant curvature.
Abstract: Introduction. Sasakian manifold with Riemannian metric is defined by S. Sasaki [5]. In this paper, we want to define Sasakian manifold with pseudo-Riemannian metric, and discuss the classification of Sasakian manifolds. In section 1, we define a Sasakian manifold (with pseudo-Riemannian metric). In section 2, we define the model spaces of Sasakian manifolds which are used in section 4 for the classification of Sasakian manifolds of constant φ-sectional curvatures. In section 3, we discuss Z)-homothetic deformation which is defined by S. Tanno [9], and prove some fundamental lemmas concerning completeness of the deformed metric. In section 5, we prove that a Sasakian manifold, satisfying R(X, Y) R = 0 for all tangent vectors X and Y, is of constant curvature. In section 6, we discuss a Sasakian manifold M^ which is properly and isometrically immersed in £f.
139 citations