Generalized Cross Product in (2+s)-Dimensional Framed Metric Manifolds with Application to Legendre Curves
Abstract
This study generalizes the cross product defined in 3-dimensional almost contact metric manifolds and describes a new generalized cross product for n=1 in (2n+s)-dimensional framed metric manifolds. Moreover, it studies some of the proposed product’s basic properties. It also performs characterizations of the curvature of a Legendre curve on an S-manifold and calculates the curvature of a Legendre curve. Furthermore, it shows that Legendre curves are also biharmonic curves. Next, this study observes that a Legendre curve of osculating order 5 on S-manifolds is imbedded in the 3-dimensional K-contact space. Lastly, the current paper discusses the need for further research.
Keywords
Generalised cross product, S-manifolds, Legendre curves
Supporting Institution
This study is supported by the Office of Scientific Resarch Projects Coordination at Çanakkale Onsekiz Mart University
Project Number
FDK-2021-3520
References
- K. Yano, On Structures Defined by a Tensor Field f of Type (1,1) Satisfying f^3+f=0, Tensor 14 (1963) 99–109.
- S. I. Goldberg, K. Yano, On Normal Globally Framed f-manifolds, Tohoku Mathematical Journal 22 (1970) 362–370.
- D. E. Blair, Geometry of Manifolds with Structural Group U(n)×O(s), Journal of Differential Geometry 4 (1970) 155–167.
- A. Sarkar, S. K. Hui, M. Sen, A Study on Legendre Curves in 3-Dimensional Trans-Sasakian Manifolds, Lobachevski Journal of Mathematics 35 (2014) 11–18.
- C. Özgür, Ş. Güvenç, On Biharmonic Legendre Curves in S-space Forms, Turkish Journal of Mathematics 38 (2014) 454–461.
- Ç. Camcı, Extended Cross Product in a 3-dimensional Almost Contact Metric Manifold with Applications to Curve Theory, Turkish Journal of Mathematics 36 (2012) 305–318.
- M. Z. Williams, F. M. Stein, A Triple Product of Vectors in Four-space, Mathematics Magazine 4 (1964) 230–235.
- S. Ishihara, Normal Structure f Satisfying f^3+f=0, Mathematical Seminar Reports 18 (1966) 36–47
- J. S. Kim, M. K. Dwivedi, M. M. Tripathi, Ricci Curvature of Integral Submanifolds of an S-space Form, Bulletin of the Korean Mathematical Society 44 (2007) 395–406.
- I. Hasegawa, Y. Okuyama, T. Abe, On p-th Sasakian Manifolds, Journal of Hokkaido University of Education (Section II A) 37 (1) (1986) 1–16.