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On kinematics and differential geometry of Euclidean submanifolds

Yılmaz Tunçer, Yusuf Yaylı, M Sa˘
2008 Balkan Journal of Geometry and Its Applications   unpublished
In this study, we derive the equations of a motion model of two smooth homothetic along pole curves submanifolds M and N ; the curves are trajectories of instantaneous rotation centers at the contact points of these submanifolds. We comment on the homothetic motions, which assume sliding and rolling. M.S.C. 2000: 53A17, 53A05, 53A04. In ([7]), R.Müller has generalized 1-parameter motions in n-dimensional Euclidean space given the equation of the form Y = AX + C, and has investigated the
more » ... tigated the so-called axoid surfaces. Further, K. Nomizu has defined ([8]) the 1-parameter motion model along the pole curves on the tangent plane of the sphere, by using parallel vector fields, and has obtained results in the particular cases of sliding and rolling. Then, H.H.Hacısaliho˘ glu has the investigated 1-parameter homothetic motion in the n-dimensional Euclidean space ([4]), and B. Karaka¸sKaraka¸s has adapted K. Nomizu's kine-matic model to homothetic motion, defining as well parallel vector fields along curves ([5]). In this study, we define the kinematic model of smooth submanifolds M and N using arbitrary orthonormal frames along the pole curves and obtain the equations of this homothetic motion which assume both rolling and sliding of M upon N along these curves. Further, we obtain the equations of homothetic motion of M on N for two given arbitrary curves on M and N respectively, assuming that these curves are pole curves. §2. Introduction We shall use hereafter the definitions and notations from ([5]). The homothetic motion of smooth submanifolds M onto N in Euclidean 3-space is generated by the transformation *