Abstract
The aim of this paper is to study (spacelike) slant curves of three-dimensional normal almost paracontact manifolds as natural generalization of Legendre curves. Such a curve is characterized by means of the scalar product between its normal vector field and the characteristic vector field of the ambient space. In the particular case of a helix, we show that it has a proper (non-harmonic) mean curvature vector field. The general expressions of the curvature and torsion of these curves and the associated Lancret invariant are computed as well as the corresponding variants for the quasi-para-Sasakian case. Two examples (one of us and one of Joanna Wełyczko) are discussed for a normal not quasi-para-Sasakian 3-manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ali AT, López R (2011) Slant helices in Minkowski space \(E^3_1\). J Korean Math Soc 48(1):159–167 [MR2778006 (2012b:53094)]
Baikoussis Ch, Blair DE (1994) On Legendre curves in contact 3-manifolds. Geom Dedicata 49(2):135–142 [MR1266269 (95a:53048)]
Barros M (1997) General helices and a theorem of Lancret. Proc Am Math Soc 125(5):1503–1509 [MR1363411 (97g:53066)]
Bejan C-L, Crasmareanu M (2014) Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry. Ann Global Anal Geom 46(2):117–127 (MR3239277)
Belkhelfa M, Hirica I-E, Rosca R, Verstraelen L (2002) On Legendre curves in Riemannian and Lorentzian Sasaki spaces. Soochow J Math 28(1):81–91 (MR1893607) (2003f:53063)
Blaga AM, Crasmareanu M (2015) Special connections in almost paracontact metric geometry, Bull. Iran Math Soc 41(6):1345–1353 (MR3437147)
Blair DE, Dillen F, Verstraelen L, Vrancken L (1997) Deformations of Legendre curves. Note Mat 15(1):99–110 [MR1611801 (99b:53076)]
Camci C, Yayli Y, Hacisalihoglu HH (2008) On the characterization of spherical curves in 3-dimensional Sasakian spaces. J Math Anal Appl 342(2):1151–1159 [MR2445265 (2009k:53097)]
Calvaruso G (2012) Homogeneous paracontact metric three-manifolds. Ill J Math 55(2):697–718 (MR3020703)
Calvaruso G (2012) Three-dimensional paracontact Walker structures. Boll Unione Mat Ital 5(2):387–403 (MR2977255)
Cappelletti Montano B (2010) Bi-paracontact structures and Legendre foliations. Kodai Math J 33(3):473–512 [MR2754333 (2012f:53170)]
Călin C, Crasmareanu M (2013) Slant curves and particles in 3-dimensional warped products and their Lancret invariants. Bull Aust Math Soc 88(1):128–142 (MR3096876)
Călin C, Crasmareanu M (2013) Slant curves in 3-dimensional normal almost contact metric geometry. Mediterr J Math 10(2):1067–1077 (MR3045696)
Călin C, Crasmareanu M (2014) Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry. Czechoslov Math J 64(4):945–960 (MR3304790)
Călin C, Crasmareanu M (2016) Magnetic curves in 3-dimensional quasi-para-Sasakian geometry. Mediterr J Math 13(4):2087–2097 (MR3530919)
Călin C, Crasmareanu M, Munteanu M-I (2012) Slant curves in 3-dimensional f-Kenmotsu manifolds. J Math Anal Appl 394(1):400–407 (MR2926230)
Cho JT, Inoguchi J-I, Lee J-E (2006) On slant curves in Sasakian 3-manifolds. Bull Aust Math Soc 74(3):359–367 [MR2273746 (2007g:53059)]
Cho JT; Lee JE (2008) Slant curves in contact pseudo-Hermitian 3-manifolds. Bull Aust Math Soc 78(3):383–396 [MR2472274 (2009m:53135)]
Choi JH, Kim YH, Ali AT (2012) Some associated curves of Frenet non-lightlike curves in \(E^3_1\). J Math Anal Appl 394(2):712–723 (MR2927492)
Crasmareanu M, Pişcoran L-I (2015) Invariant distributions and holomorphic vector fields in paracontact geometry. Turk J Math 39(4):467–476 (MR3366733)
Dacko P, Olszak Z (2007) On weakly para-cosymplectic manifolds of dimension \(3\). J Geom Phys 57(2):561–570 [MR2271205 (2008e:53038)]
Fetcu D (2008) Biharmonic Legendre curves in Sasakian space forms. J Korean Math Soc 45(2):393–404 [MR2389544 (2008m:53159)]
Ivanov S, Vassilev D, Zamkovoy S (2010) Conformal paracontact curvature and the local flatness theorem. Geom Dedicata 144:79–100 [MR2580419 (2011b:53174)]
Kaneyuki S, Williams FL (1985) Almost paracontact and parahodge structures on manifolds. Nagoya Math J 99:173–187 [MR0805088 (87a:53071)]
Kühnel W (2002) Differential geometry. Curves-surfaces-manifolds. Translated from the 1999 German original by Bruce Hunt. Student Mathematical Library, vol 16. American Mathematical Society, Providence, RI [MR1882174 (2002k:53001)]
Lee J-E, On Legendre curves in contact pseudo-Hermitian 3-manifolds. Bull Aust Math Soc 81(1):156–164 [MR2584930 (2011a:53096)]
Özgür C, Güvenç S (2012) On some types of slant curves in contact pseudo-Hermitian 3-manifolds. Ann Polon Math 104(3):217–228 (MR2914532)
Smoczyk K (2003) Closed Legendre geodesics in Sasaki manifolds. N Y J Math 9:23–47 [(electronic) MR2016178 (2004j:53085)]
Welyczko J (2007) On Legendre curves in 3-dimensional normal almost contact metric manifolds. Soochow J Math 33(4):929–937 [MR2404614 (2009d:53119)]
Welyczko J (2009) On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Results Math 54(3-4):377–387 [MR2534454 (2010g:53153)]
Welyczko J (2014) Slant curves in 3-dimensional normal almost paracontact metric manifolds. Mediterr J Math 11(3):965–978 (MR3232573)
Zamkovoy S (2008) Canonical connections on paracontact manifolds. Ann Global Anal Geom 36(1):37–60 [MR2520029 (2010d:53029)]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crasmareanu, M., Frigioiu, C. Space-Like Slant Curves in Three-Dimensional Normal Almost Paracontact Geometry. Iran J Sci Technol Trans Sci 41, 1123–1129 (2017). https://doi.org/10.1007/s40995-017-0232-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0232-y
Keywords
- Normal almost paracontact manifold
- (spacelike) slant
- Legendre curve
- Lancret invariant
- (generalized) helix