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PAIRED CURVES ON RIEMANNIAN MANIFOLDS

VISHESH , S BHAT (2019) PAIRED CURVES ON RIEMANNIAN MANIFOLDS. PhD thesis, CHRIST(Deemed to be University).

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Abstract

The thesis examines certain curves in Riemannian manifolds, which exist in one-one correspondence with other curves. This correspondence is described by means of a rigid link between the frame vectors allied to these "paired" curves. The Combescure transformation is made use of to exhibit that one of the paired curves can be obtained as an infinitesimal deformation of the other. The correspondence between the paired curves is exploited to formulate expressions relating the curvature and torsion functions of one with the other. The primary setting for this study is the Euclidean space R3, with brief considerations of 3-dimensional Riemannian space forms (i.e. Riemannian manifolds of constant sectional curvature). The exponential map is the desired tool of analysis in the latter case. The major classes of curves treated are the Mannheim class of curves in R3 and their partner curves. Further, with the help of the notion of the osculating helix, a new class of curves called constantpitch curves are defined, which are seen to naturally arise in motion analysis studies in theoretical kinematics. Constant-pitch curves are also shown to be inherent counterparts of Mannheim curves by means of a deformation. Pivotal properties of Mannheim and constant-pitch curves are established and a few examples are put forth. Integral characterizations of both curves are derived in terms of their spherical indicatrices. A consequence of this to geometric modeling problems involving energy functionals and also to the study of elastic curves is discussed. Certain ruled surfaces generated by Mannheim and constant-pitch curves which occur as axodes associated to a rigid body motion are detailed and their applications to kinematics are studied. Further, the nature of paired curves in connection with tubular neighbourhoods/surfaces are investigated.

Item Type:Thesis (PhD)
Subjects:Thesis > Ph.D > Mathematics
Thesis
Thesis > Ph.D
ID Code:7792
Deposited By:Shaiju M C
Deposited On:13 May 2019 09:23
Last Modified:28 May 2019 12:49

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