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SUPERVISOR
Azam Etemad dehkordy,Mansour Aghasi
اعظم اعتماددهکردی (استاد راهنما) منصور آقاسی (استاد مشاور)
 
STUDENT
Samira Khani
سمیرا خانی

FACULTY - DEPARTMENT

دانشکده ریاضی
DEGREE
Master of Science (MSc)
YEAR
1390

TITLE

Generalized Rotation Surfaces in E^4
In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. So, we give some basic concepts of the surfaces in E4 and some special examples of these surfaces in E4. Further, the curvature properties of these surfaces are investigated. We consider Frenet curve with constant curvature. Since these curves are trajectories of the 1-parameter group of the Euclidean transformations, so, Klein and lie called them W-curves.We explain some geometric properties of W-curves. In 2008 Ganchev and Milousheva considered the surface M generated by a W-curve in a Euclidean 4-space. This curve is a generalization of the circular helix in an Euclidean space of three dimensions. They have shown that these generated surfaces are a special type of rotation surfaces which are introduced first by C.Moore in 1919. We give a necessary and sufficient condition for a W-curve on a trous to be twisted. We consider ruled surfaces imbedded in a Euclidean space of four dimensions. In 1936 Plass studied ruled surfaces imbedded in E4.In 1980 Rouxel considered ruled Chen surfaces in E4. We also calculate the Gaussian and mean curvature vector of generalized rotation surfaces and give a necessary and sufficient condition for vanishing Gaussian and mean curvature. We define Vranceanu rotation surface and Clifford torus and tensor product surface of two Euclidean planer curves in a Euclidean 4-space. We show that tensor product surface of two Euclidean planer curves is a minimal surface in E4 if and only if one of curves is a straight line through 0. In 1973 Chen defined the allied vector field a(v) of a normal vector field v. In particular, the allied mean curvature vector field is orthogonal to H. Further, B.Y.Chen defined the A -surface to be the surface for which a(H) vanishes identically. Such surfaces are also called Chen surfaces. The normal; MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none" surfaces contain all minimal and pseud-umbilical surfaces, and also all surfaces for which dim N 1 1, in particular all hypersurfaces. We also show that every general rotation surface is Chen surface in E4. We give a necessary and sufficient condition for generalized rotation surfaces to become pseudoumbilical. We prove that each minimal surface is pseudo-umbilical. Further, we present some examples of the generalized rotation surfaces in E4. It is show that they are all non-trivial Chen surfaces. Such as Blashke, Klein bottle, Banchoff and Lawson surface. Finally we define normal section and curvature ellipse of a smooth surface. The characterization of point in the surfaces as elliptic, parabolic and hyperbolic point, and the inflection point, are also discussed. We calculate curvature ellipse of Chen surface in a Euclidean space of four dimensions. We prove that if a non-trivial Chen surface in E4 is a flat surface then the origin of the normal plane T p M lies outside of the curvature ellipse of Chen surface. Further, we give the necessary and sufficient condition for the origin of T P M to lie on the curvature ellipse of Vranceanu surfaces, which are special type of Chen surfaces.
این پایان نامه مشتمل بر تعمیم نظریه‌ی رویه‌های دوار از فضای اقلیدسی سه بعدی به فضای اقلیدسی چهار بعدی است. بنابراین لازم است که تعریف مناسبی از این رویه‌ها در E^4 ارائه شود. از جمله ابزار بکار رفته در این تعریف‌ها مبحث W-خم‌ها که معرفی می‌گردد و مثال‌هایی برای آن مطرح می‌شود. همچنین نوع خاصی از رویه‌های دوار تعمیم یافته با نام ورونیس ارائه می‌شود. سپس انحنای گاوسی و انحنای میانگین آن محاسبه خواهد شد. در ادامه اثبات می‌شود که هر رویه‌ی دوار تعمیم یافته در E^4 رویه‌ی چن است. بعلاوه چند مثال از رویه‌های دوار تعمیم یافته در فضای اقلیدسی چهار بعدی جهت ملموس کردن مطالب مورد نظر در ادامه خواهد آمد. در انتها به محاسبه‌ی بیضی انحنای رویه‌های چن پرداخته می‌شود.

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