Pseudo-Hermitian magnetic curves in normal almost contact metric 3-manifolds
2
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric $3$-manifold equipped with the canonical affine connection $\hat\nabla^t$ is a slant helix with pseudo-Hermitian curvature $\hat{\kappa}=|q|\sin\theta$ and pseudo-Hermitian torsion $\hat\tau=q\cos\theta$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric $3$-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve $\gamma$ in a quasi-Sasakian 3-manifold $M$ is a slant curve with curvature $\kappa=|(t-\alpha)\cos\theta\>+q|\sin\theta$ and torsion $\tau=\alpha+\{(t-\alpha)\cos\theta+q\}\cos\theta$. These curves are not helices, in general. Note that if the ambient space $M$ is an $\alpha$-Sasakian $3$-manifold, then $\gamma$ is a slant helix.
Abstract
In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric $3$-manifold equipped with the canonical affine connection $\hat\nabla^t$ is a slant helix with pseudo-Hermitian curvature $\hat{\kappa}=|q|\sin\theta$ and pseudo-Hermitian torsion $\hat\tau=q\cos\theta$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric $3$-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve $\gamma$ in a quasi-Sasakian 3-manifold $M$ is a slant curve with curvature $\kappa=|(t-\alpha)\cos\theta\>+q|\sin\theta$ and torsion $\tau=\alpha+\{(t-\alpha)\cos\theta+q\}\cos\theta$. These curves are not helices, in general. Note that if the ambient space $M$ is an $\alpha$-Sasakian $3$-manifold, then $\gamma$ is a slant helix.
참고문헌 (14)
- D. E. Blair, "Riemannian geometry of contact and symplectic manifolds" (2002), Birkh¨auser Boston, Inc.
- J. T. Cho, "On some classes of almost contact metric manifolds", Tsukuba J. Math, 19(1), pp.201~217 (1995)[DOI]
- J. T. Cho, "On slant curves in Sasakian 3-manifolds", Bull. Austral. Math. Soc, 74(3), pp.359~367 (2006)[DOI]
- J. T. Cho, "Biharmonic curves in 3-dimensional Sasakian space forms", Ann. Mat. Pura Appl. (4), 186(4), pp.685~701 (2007)[DOI]
- S. -L. Drut¸˘a-Romaniuc, "Magnetic curves in cosymplectic manifolds", Rep. Math. Phys, 78(1), pp.33~48 (2016)[DOI]
- S. G¨uven¸c, "On pseudo-Hermitian magnetic curves in Sasakian manifolds", arXiv:2002.05214v1, ~ (2020)
- O. Ikawa, "Motion of charged particles in Sasakian manifolds", SUT J. Math, 43(2), pp.263~266 (2007)
- J. Inoguchi, "Almost contact curves in normal almost contact 3-manifolds", J. Geom, 103(3), pp.457~474 (2012)[DOI]
- J. Inoguchi, "Affine biharmonic curves in 3-dimensional homogeneous geometries", Mediterr. J. Math, 10(1), pp.571~592 (2013)[DOI]
- J. Inoguchi, "Magnetic curves in quasi-Sasakian 3-manifolds", Anal. Math. Phys, 9(1), pp.43~61 (2019)[DOI]
- M. Okumura, "Some remarks on space with a certain contact structure", Tohoku Math. J. (2), 14, pp.135~145 (1962)[DOI]
- Z. Olszak, "Normal almost contact metric manifolds of dimension three", Ann. Polon. Math, 47(1), pp.41~50 (1986)[DOI]
- S. Sasaki, "On differentiable manifolds with certain structures which are closely related to almost contact structure. II", Tohoku Math. J. (2), 13, pp.281~294 (1961)[DOI]
- S. Tanno, "Variational problems on contact Riemannian manifolds", Trans. Amer. Math. Soc, 314(1), pp.349~379 (1989)[DOI]