Paired curves on riemannian manifolds

Title

Paired curves on riemannian manifolds

Creator

Bhat, Vishesh S.

Contributor

Baskar, Hari R

Description

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 newlineto these quotpairedquot 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. newlineThe 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 newlinenotion 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 newlinealso shown to be inherent counterparts of Mannheim curves by means of a deformation. newlinePivotal 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 newlineconsequence of this to geometric modeling problems involving energy functionals and also to the study of elastic curves is discussed. newlineCertain ruled surfaces generated by Mannheim and constant-pitch curves which occur as axodes associated to a rigid body motion are newlinedetailed and their applications to kinematics are studied. Further, the nature of paired curves in connection with tubular neighbourhoods/surfaces are investigated. newline

Source

Author's Submission

Date

2019-01-01

Publisher

Christ(Deemed to be University)

Subject

Mathematics and Statistics

Rights

Open Access

Relation

61000105

Format

PDF

Language

English

Type

PhD

Identifier

http://hdl.handle.net/10603/244156