Vanlı A.
American Mathematical Society, ss.1, Michigan, 2021
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Yayın Türü:
Bilirkişi Raporu / Uzmanlık Raporu
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Basım Tarihi:
2021
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Basıldığı Şehir:
Michigan
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Açık Arşiv Koleksiyonu:
AVESİS Açık Erişim Koleksiyonu
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Gazi Üniversitesi Adresli:
Evet
Özet
`In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit `good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α,δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behavior under a new class of deformations, called H-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α,δ)-Sasaki manifold is Einstein either if α=δ (the 3-α-Sasaki case) or if δ=(2n+3)α, where dimM=4n+3.
``In the second part we find these adapted connections. We start with a very general notion of ϕ-compatible connections, where ϕ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α,δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.''