The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

AbstractRecently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of H-harmonic and H*-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

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A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-060-x ◽

1992 ◽

Vol 35 (4) ◽

pp. 455-462 ◽

Cited By ~ 7

Author(s):

D. E. Blair ◽

D. Perrone

Keyword(s):

Scalar Curvature ◽

Critical Point ◽

Integral Functional ◽

Riemannian Metrics ◽

Contact Manifold ◽

Contact Condition ◽

Contact Metric Manifolds ◽

Point Condition ◽

Webster Scalar Curvature ◽

Group Of Isometries

AbstractChern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature.

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A remark on transformations of a $K$-contact manifold

Tohoku Mathematical Journal ◽

10.2748/tmj/1178243702 ◽

1964 ◽

Vol 16 (2) ◽

pp. 173-175 ◽

Cited By ~ 1

Author(s):

Shûkichi Tanno

Keyword(s):

Contact Manifold

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Applications of Square Roots of Diffeomorphisms

Axioms ◽

10.3390/axioms8020043 ◽

2019 ◽

Vol 8 (2) ◽

pp. 43

Author(s):

Yoshihiro Sugimoto

Keyword(s):

Smooth Manifold ◽

Contact Manifold ◽

Diffeomorphism Group ◽

Square Root ◽

Square Roots ◽

Group Diff

In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C ∞ -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C ∞ -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) ∖ Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.

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Ricci almost solitons and contact geometry

Advances in Geometry ◽

10.1515/advgeom-2019-0026 ◽

2019 ◽

Vol 0 (0) ◽

Cited By ~ 1

Author(s):

Amalendu Ghosh

Keyword(s):

Vector Field ◽

Contact Manifold ◽

Ricci Soliton ◽

Jacobi Field ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Rigidity Results ◽

Parallel Ricci Tensor ◽

Potential Vector Field

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

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