Geometry Of Pair Of Curves In Space Forms

In differential geometry,the theory of curves in Euclidean space and Minkowski space is one of the main research areas.In the theory of curves,involute-evolute curve,Bertrand curve,helices,slant helices,and rectifying curves are the curves of great interest.Related curves of a specific curve are also investigated.Among these curves,the most considered ones are involute-evolute curve,Bertrand curves,spherical indicatrices and Mannheim partner curves.Involute-evolute curves are the most interesting and fascinating curves in classical differ-ential geometry.The involute of a given curve is a well-known concept in Euclidean space as well as in Minkowski space.A new curve that is based on a given curve can be derived from Involvents that plays a significant role in construction of gears.This dissertation explores involute-evolute pair of curves for some spaces.This dissertation comprises of the following topics.(1)In classical differential geometry,the theory of involute-evolute curves in 4-dimensional Euclidean space is one of the main study areas.Evolute curves were studied by some researchers in 4 dimensional Euclidean space.However the special characters of the curve are not considered which is a research gap in this technique.This dissertation proposes some terminologies for generalized involute and evolute curve-couple in 4-dimensional Euclidean space.At any point of the curve ?,the plane spanned by ?T,B1? is called the(0,2)-tangent plane of curve ? and the plane spanned by ?N,B3?is called the(1,3)-normal plane of ?.In this study a pair of involute and evolute curve is considered in 4-dimensional Euclidean space.This dissertation provides a generalized involute and evolute curve-couple in 4-dimensional Euclidean space.Using proposed terminologies,we obtained significant results which can be more beneficial for researchers in future studies.(2)Extensive research has been done on evolute curves in Minkowski space-time.However,the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature.In this thsis,we have generalized a special kind of evolute and involute curve in four-dimensional Minkowski space.We introduced some methodologies for proposed study.We consider(1,3)-evolute curves with respect to the casual characteristics of the(1,3)-normal plane that is spanned by the principal normal and the second binormal of the vector fields and the(0,2)-evolute curve that is spanned by the tangent and first binormal of the given curve.We restrict our investigation of(1,3)-evolute curves to the(1,3)-normal plane in four-dimensional Minkowski space.Using proposed methodologies,this research contribution obtains a necessary and sufficient condition for the curve possessing the generalized evolute as well as the involute curve.Furthermore,the Cartan null curve is also discussed in detail.The understanding of evolute curves with this new type of generalized evolute and involute curve in four-dimensional Minkowski space will be more beneficial for mathematicians in future studies.(3)The theory of curves has been one of the exciting subject because of having many application area from geometry to the different branch of science.In differential gometry,there are many important consequences and properties in the theory of the curves.In this dissertation,we define an involute of order k of a null Cartan curve in Minkowski space E1n for n?3 and 1?k?n-1.As regards,the study demonstrate that if a null Cartan helix has a null Cartan involute of order 1 or 2,then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve.The study also show that among all null Cartan curves in E13,only the null Cartan cubic has two families of involutes of order 1,one of which lies on B-scroll.The dissertation also provides some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space.As an application,the study show that involutes of order 1 of a null Cartan curve in E13,evolving according to null Betchov-Da Rios vortex filament equation,generate timelike Hasimoto surfaces.