N(κ)-contact metric manifolds admitting Z-tensor

Year 2020, Volume: 2 Issue: 1, 64 - 69, 29.12.2020

İnan Ünal

Abstract

Contact manifolds have many applications to medical science, technology, geometric optics, geometric quantization, control theory, thermodynamics, and classical mechanics. Therefore, studies on the Riemann geometry of contact manifolds are important. On the other hand, one of the important tools of Riemann geometry are curvature tensors. By using curvature tensors, some geometric properties and physical applications of contact manifolds can be examined. Especially, important results are obtained on symmetry of manifolds which is the one of the main study topics of Riemannian geometry. A special kind of curvature tensor is the (0,2) -type $\mathcal{Z}$-tensor which has some geometric properties different from Ricci curvature tensor. This type of tensor gives us important results on contact manifolds. Especially, semi-symmetry conditions which are related to $\mathcal{Z}$-tensor present nice results. In this study, we work on$N(k)$- contact metric manifolds which are a special kind of contact manifolds. We present some results on $N(k)$-contact metric manifolds by using$\mathcal{Z}$-tensor. We classify the manifolds by using some semi-symetry conditions such as $R(\xi ,W).\mathcal{Z}=0$, $\mathcal{P}(\xi ,W).\mathcal{Z}=0$, $\mathcal{L}(\xi ,W).\mathcal{Z}=0$ and \[{{\mathcal{W}}_{2}}(\xi ,W).\mathcal{Z}=0\], where R the Riemann curvature tensor, $\mathcal{P}$is the Projective curvature tensor, $\mathcal{L}$ is the concircular curvature tensor and \[{{\mathcal{W}}_{2}}\]is the $W_2$ curvature tensor.

Keywords

N(k)-contact metric manifold, , Z-Tensor,, semi-seymmetry

N(κ)-Contact Metric Manifolds Admitting Z-Tensor

Abstract

Kontakt manifoldların tıp bilimi, teknoloji, geometrik optik, geometrik nicemleme, kontrol teorisi, termodinamik ve klasik mekanikte birçok uygulaması vardır. Bu nedenle, temas manifoldlarının Riemann geometrisi üzerine yapılan çalışmalar önemlidir. Eğrilik tensörleri Riemann geometrisinin önemli araçlarından biridir. Eğrilik tensörleri kullanılarak, kontakt manifoldların bazı geometrik özellikleri ve fiziksel uygulamaları incelenebilir. Özellikle, Riemann geometrisinin ana çalışma konularından biri olan simetri özellikleri hakkında önemli sonuçlar elde edilmektedir. Önemli eğrilik tensörlerinden biri olan $\mathcal{Z}$-tensör , Ricci tensöründen farklı bazı geometrik özelliklere sahip (0,2) -tipindeki bir tensördür. $\mathcal{Z}$-tensörü, kontakt manifoldlarda bize önemli sonuçlar verir. Özellikle, $\mathcal{Z}$-tensör ile ilişkili yarı-simetri koşulları üzerine önemli sonuçlar elde edilmektedir. Bu çalışmada, manifoldları olan temaslı metrik manifoldlar üzerinde çalışılmıştır. R the Riemann eğrilik tensörü, $\mathcal{P}$is the projectif eğrilik tenrösü, $\mathcal{L}$ is the concircular eğrilik tensörü and \[{{\mathcal{W}}_{2}}\]is the $W_2$ eğrilik tensörü olmak üzere, . manifoldlar $R(\xi ,W).\mathcal{Z}=0$, $\mathcal{P}(\xi ,W).\mathcal{Z}=0$, $\mathcal{L}(\xi ,W).\mathcal{Z}=0$ and \[{{\mathcal{W}}_{2}}(\xi ,W).\mathcal{Z}=0\] gibi yarı-simetri şartları altında incelenmiştir.

Keywords

N(k)-kontakt metric manifold, Z-tensör, yarı-simetri

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