A curve theory on sliced almost contact manifolds

Yıl 2022, , 62 - 76, 30.04.2022

Mehmet Gümüş , Çetin Camcı

https://doi.org/10.54187/jnrs.1095343

Öz

We have realized a gap between almost contact metric manifolds and contact metric manifolds in our studies. The examples that were given as Sasaki manifolds don't satisfy the condition of being contact metric manifold. As a result of our work, the sliced almost contact manifolds were formed and defined in \cite{MG}. In this paper we applied the theory of sliced almost contact manifolds to curves as a curve theory in three dimensional space. We define the $\pi-regular$ and $\pi-Legendre$ curves, also we give basic theorems on $\pi-Legendre$ curves and an example to $\pi-Legendre$ curves.

Anahtar Kelimeler

Sliced almost contact manifolds, Sliced contact metric manifolds, π-Legendre curve

Kaynakça

• M. Gümüş, A new construction of Sasaki manifolds in semi-Riemann space and applications, PhD Dissertation, Çanakkale Onsekiz Mart University (2018) Çanakkale, Turkey (in Turkish).
• C. Huygens, The pendulum clock or geometrical demonstrations concerning the motion of pendula as applied to clocks, Iowa State University Press, Iowa, 1986.
• J. M. Child, The geometrical lectures of Isaac Barrow, The Open Court Publishing Company, 1916.
• S. I. Newton, The mathematical principles of natural philosophy: Philosophiae naturalis principia mathematica, CreateSpace Independent Publishing Platform, 2016.
• S. Lie, Verallgemeinerung und neue Verwertung der Jacobischen Multiplikatortheorie, Forhandlinger Christiania, (1874) 255-274.
• J. Gray, Some global properties of contact structures, Annals of Mathematics, 69, (1959) 421-450.
• J. W. Gibbs, A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, Transactions of the Connecticut Academy of Arts and Sciences, 2, (1873) 382-404.
• B. O'Neill, Semi-Riemannian geometry. with applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc.[Harcourt Brace Jovanovich, Publishers], New York, 1983.
• S. Sasaki, Y. Hatakeyama, On differentiable manifolds with contact metric structures, Journal of the Mathematical Society of Japan, 14, (1962) 249-271.
• D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Springer-Verlag, 509, 1976.
• C. Baikoussis, D. E. Blair, On Legendre curves in contact 3-manifolds, Geometriae Dedicata, 49, (1994) 135-142.
• K. L. Duggal, A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Kluwer Academic, 364, 1996.
• M. Belkhelfa, I. E. Hiric, R. Rosca, L. Verstraelen, On Legendre curves in Riemannian and Lorentzian Sasaki spaces, Soochow Journal of Mathematics, 28, (2002) 81-91.
• D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progressin Mathematics 203. Birkhauser Boston, 2002.
• Ç. Camcı, A curve theory in contact geometry, PhD Dissertation, Ankara University (2007) Ankara, Turkey (in Turkish).
• K. L. Duggal, B. Şahin, Lightlike submanifolds of indefinite Sasakian manifolds, International Journal of Mathematics and Mathematical Sciences, 2007, (2007) Article ID: 057585.
• M. Gümüş, Ç. Camcı, Riemannian curvature of a sliced contact metric manifold, Çanakkale Onsekiz Mart University, Journal of Graduate School of Natural and Applied Sciences, 4(2), (2018) 1-14.
• K. Yano, M. Kon, Structures on manifolds, World Scientific, 1984.
• Ç. Camcı, Extended cross product in 3-dimensional almost contact metric manifold with applications to curve theory, Turkish Journal of Mathematics, 36, (2012) 305-318.