Riemannian Geometry of Contact and Symplectic Manifolds

Blair, David E.

doi:10.1007/978-0-8176-4959-3

Cited by 972 publications

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“…A K-contact structure is Sasakian only in dimension 3, and this fails in higher dimensions. More details on contact manifolds can be found in [Blair 2002].…”

Section: Preliminariesmentioning

confidence: 99%

On the concircular curvature of a (κ,μ,ν)-manifold

Gouli-Andreou

Moutafi²

2014

Pacific J. Math.

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We study (κ, µ,ν)-contact metric 3-manifolds (a notion introduced by Koufogiorgos, Markellos and Papantoniou) that are Ricci flat, or are Einstein but not Sasakian, or satisfy ∇ Z = 0, where Z is the concircular curvature tensor, or satisfy Z(ξ, X)· Z = 0, where ξ is the Reeb field, or satisfy Z(ξ, X)· S = 0, where S is the Ricci tensor, or finally satisfy R(ξ, X) · Z = 0, where R is the Riemannian curvature tensor.(with κ ≤ 1 and if κ = 1 the structure is Sasakian. The full classification of these manifolds was given by E. Boeckx [2000]. If µ = 0 we have the κ-nullity distribution and if ξ ∈ N (κ) we have a N (κ)-contact metric manifold. Koufogiorgos MSC2010: primary 53C15, 53C25, 53D10; secondary 53C35.

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“…A K-contact structure is Sasakian only in dimension 3, and this fails in higher dimensions. More details on contact manifolds can be found in [Blair 2002].…”

Section: Preliminariesmentioning

confidence: 99%

On the concircular curvature of a (κ,μ,ν)-manifold

Gouli-Andreou

Moutafi²

2014

Pacific J. Math.

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“…Thus, 2 +1 (1) is equipped with an almost contact structure ( , , ). Together with the induced metric, this structure is Sasakian (see [4]). 7 (1) such that ℎ( ( ), ( )) = 0 and ℎ( ( ), ( )) = 0.…”

Section: Preliminariesmentioning

confidence: 99%

Four‐dimensional contact ‐submanifolds in

Djorić

Munteanu

Vrancken

2017

Mathematische Nachrichten

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The analogue of CR‐submanifolds in (almost) Kählerian manifolds is the concept of contact CR‐submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant with respect to the endomorphism φ and E(M) is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four‐dimensional contact CR‐submanifolds in S5false(1false) and S7false(1false) for which the second fundamental form restricted to H(M) and E(M) vanishes identically.

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“…A differentiable manifold M 2n+1 with almost contact metric structure (φ, ξ, η, g) is called almost contact metric manifold [1]. It is easy to see that the tensor φ ij , which is defined by φ Let M 2n+1 (g, ω) be a Weyl manifold with the connection D. Then M 2n+1 (g, ω) has an almost contact structure if the following conditions are satisfied in addition to the conditions (4.1)-(4.6) [8]:…”

Section: Almost Contact Weyl Manifoldsmentioning

confidence: 99%