Curves on a Smooth Surface with Position Vectors Lie in the Tangent Plane
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DOI: 10.1007/s13226-020-0452-2
CURVES ON A SMOOTH SURFACE WITH POSITION VECTORS LIE IN THE TANGENT PLANE Absos Ali Shaikh and Pinaki Ranjan Ghosh Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713 104, West Bengal, India e-mails: [email protected], [email protected] [email protected]; [email protected] (Received 8 April 2019; accepted 11 June 2019) The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length of the position vector, tangential component of the position vector and geodesic curvature of a curve on a surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces. Key words : Isometry of surfaces; first fundamental form; second fundamental form; geodesic curvature. 2010 Mathematics Subject Classification : 53A04, 53A05, 53A15.
1. I NTRODUCTION The notion of rectifying curve was introduced by Chen [3] as a curve in the Euclidean space such that its position vector always lies in the rectifying plane, and then investigated some properties of such curves. For further properties of rectifying curves, the reader can be consulted [6] and [8]. Again Ilarslan and Nesovic [7] studied the rectifying curves in Minkowski space and obtained some of its characterization. In [9] Camci et al. associated a frame different from Frenet frame to curves on a surface and deduced some characterization of its position vector. In [4] and [5] the present authors studied rectifying and osculating curves and obtained some conditions for the invariancy of such curves under isometry. Also the invariancy of the component of position vector of rectifying and osculating curves along the normal and tangent line to the surface are obtained under isometry of surfaces.
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ABSOS ALI SHAIKH AND PINAKI RANJAN GHOSH
Motivating by the above studies of curves whose position vectors are confined in some plane, in this paper we have investigated curves on a smooth surface with position vector always lying in the tangent plane of the smooth surface. By using the Gauss equation we have deduced the component of the position vector along the tangent, normal and binormal vector in simple form. By considering isometry between two smooth surfaces it is proved that curves on smooth surface whose position vectors lie in the tangent plane are invariant. It is also shown that the length of position vector, tangential component and geodesic curvature of such curves are invariant under isomerty. 2. P RELIMINARIES This section is concerned with some preliminary notions of rectifying curves, osculating curves, isometry of surfaces and geodesic curvature (for details see, [1, 2]) which will be needed for the remaining. At every point of an unit speed parametrized curve γ(s) with atleast fourth order continuous derivative, there is an orthonormal frame of three vectors, namely, tangent,
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