ﺧﻮاﺻﯽ از نگاشت ﮔﺎوس روي ﮔﻮﻧﻪ ﺟﺪﯾﺪي از روﯾﻪ ﻫﺎي ﻟﻮﻟﻪ اي ﺷﮑﻞ در ﻓﻀﺎي اﻗﻠﯿﺪﺳﯽ 4- ﺑﻌﺪي

چكيده انگليسي :

Some Properties of Gauss map on a new type of tubular surface in four dimensional Euclidean space Neda Rostami neda rostami@math iut ac ir September 20 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Azam Etemad Dehkordi ae110mat@cc iut ac irAdvisor Dr Amir Hashemi amir hashemi@cc iut ac ir2000 MSC 53C40 53C42Keywords tubular surface parallel transport frame Gauss map pointwise 1 type Abstract The original part of this thesis is the study of Gauss map on a new type of tubular surface based on 22 28 Theconcept of finite type immersions was interoduced by Chen in 1981 15 Then he writes some papers related to thistopic An isometric immersion x M E m for a submanifold M of an Euclidean space E m that also known asthe position vector field of M is called as a finite type if it is written as a finite sum of eigenvectors of the Laplacian of M for a constant map x0 and non constant maps x1 x2 xk i e x x0 k xi If the eigenvectors i 1of the Laplacian are different numbers then the submanifold is called as k type This term is extended to the Laplacianof Gauss map of M as G a G C for a real number a and a constant vector C by Chen and Piccinniin 19 The Gauss map G of a submanifold that satisfies in above relation is called of 1 type Gauss map Thereexists many papers with titles about submanifolds having 1 type Gauss map G In the sequel by replacing a witha non constant function in above equation that is G G C new researches are done A submanifoldis said to have pointwise 1 type Gauss map G if satisfies in the new equation If the function is non constant thepointwise 1 type Gauss map is called as proper Also if the vector C is zero the pointwise 1 type Gauss map iscalled as the first kind and it is called as second kind if C 0 In this thesis at first we review the Frenet fram intwo and three dimensional euclidean spaces then we study the Frenet frame for four dimensional Euclidean space The simplest curves in two three and four dimensional Euclidean spaces is the other subject that we study about inthe first chapter Because of proof of a theorem in chapter 2 we also have a review on Euler angles in the end of firstchapter In chapter 2 we introduce the parallel transport frame and we derive relations between this frame and Frenetframe Then we study normal rectifying and osculating curves according to parallel transport frame in details weobtain the sufficient and necessary conditions for unit speed curves in Euclidean 4 space to be normal rectifying andosculating curves For a space curve u a spine curve we can define a canal surface as the envelope of a one parameter family of spheres whose centers are the points of the spine curve u and whose radii r u are varying If the radius function r u is constant the canal surface is called as a tubular tube or a pipe surface Actually thenotion of a canal surface is a generalization of an offset of a plane curve Finally in the third chapter of this thesis weconsider tubular surfaces in four dimensional Euclidean space Initially we generalize the basic concepts of classicaldifferntial geomtry to 4 dimensional Euclidean space Then we study tubular surfaces having pointwise 1 type Gaussmap in Euclidean 4 space E4 Brie y it is proved that there is no tubular surface having harmonic Gauss map Wealso obtain a spatial classification for tubular surface due to pointwise 1 type Gauss map

رستمي شاپورآبادي، ندا

ﺧﻮاﺻﯽ از نگاشت ﮔﺎوس روي ﮔﻮﻧﻪ ﺟﺪﯾﺪي از روﯾﻪ ﻫﺎي ﻟﻮﻟﻪ اي ﺷﮑﻞ در ﻓﻀﺎي اﻗﻠﯿﺪﺳﯽ 4- ﺑﻌﺪي

لوله اي شكل , خم هاي موازي , نقطه وار نوع اول , نگاشت گاوس