ناورداها و قضيه اي از نوع بونه براي رويه ها در فضاي چهار بعدي اقليدسي

چكيده انگليسي :

Invariants and Bonnet type Theorem for Surfaces in Four Dimensional Euclidean Space Abstract The local theory of surfaces in was mainly created by Gaspard Monge and Leonard Euler in the 18th century Carl Friedrich Gauss was the first mathematician who studied about intrinsic geometry of parametrized surfaces in He began to study the unit normal vector field changes to survey of geometric properties of surfaces Later it was specified there exist surfaces such that they have not any isometric immersion in and therefore need to consider parametrized surfaces in At the early of 20th century the local theory of surfaces in continued by a number of mathematicians like Eisenhart Cartan Struik Schouten Wilson Moore and Kommerell In because the surface has two normal vector so we should use of the method of tensor calculus But it had not been invented yet at the beginning of the 20th century so the called mathematicians for studying surfaces in defined an specific structure in the normal space of surface which is named ellipse of normal curvature and geometric properties of surface obtain of geometric properties ellipse properties This Thesis mainly is based on the studies of two mathematicians Ganchev and Milousheva In this way we considered parametrized surfaces in In the tangent space at any point of surface as the classical case we introduced the linear map called wiengarten map It was proved that this map is a geometric invariant of surfaces Using this map conclude two new invariant and and based on these two quantities the points on the surface are devied into four types flat elliptic hyperbolic parabolic As the classical case in each point of tangent plane of surface we can define the bilinear form named second fundamental form and it is proved that this bilinear form is a geometric invariant of surface Afterward similar to classical case principle directions and asymptotic directions are defined and proved that the relation between the number of these directions in each point of surface and the type of that point are exactly like classical cases Similar to the Dupin indicatrix in the classical case in the tangent space at any point of surface we can define a conic named tangent indicatrix It is also prove that the relation between the shape of tangent indicatrix and the type of a point flat elliptic of surface is exactly similar to the relation between the shape of Dupin ndicatrix and the type of a point in the classical case Here it is studied on the introduction of another concept called ellipse of normal curvature and described the relation between figur of tangent and the figur of their tangent indicatrix and the figur of their ellipse of normal curvature are indicatrix and the ellipse of normal curvature and the type of the point flat elliptic hyperbolic and parabolic Minimal surfaces and surfaces with flat normal connection are characterized by new invariant of surface studied By choosing an unique orthogonal frame field in each point of surface is found eight The fundamental theorem of local theory of surface in states that if six smooth functions PDF created with pdfFactory trial version www pdffactory com

مهر محمدي، منصور

ناورداها و قضيه اي از نوع بونه براي رويه ها در فضاي چهار بعدي اقليدسي

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