Ders Bilgileri

Ders Tanımı

Ders Kodu Yarıyıl T+U Saat Kredi AKTS
ABC MAT 007 0 3 + 0 3 6
Ön Koşul Dersleri Students are assumed to be familiar with Differential Geometry I, II and Advances Differential Geometry.
 Dersin Dili Türkçe Dersin Seviyesi Doktora Dersin Türü ZORUNLU Dersin Koordinatörü Prof.Dr. MEHMET ALİ GÜNGÖR Dersi Verenler Prof.Dr. MEHMET ALİ GÜNGÖR Dersin Yardımcıları Dersin Kategorisi Dersin Amacı The Differential Geometry of Manifolds course aims to give the fundamental knowledge for the studies of graduate students who study at geometry branch. Dersin İçeriği Riemannian manifolds, Covariant differentiation, Curvature tensor, Theorem of Frobenius, induces connection and second fundamental form, Equation of Gauss, Codazzi and Ricci, Scalar curvature of submanifolds, minimal submanifolds in euclidean space, minimal submanifolds of a submaniforld, Examples of minimal submanifolds, Surfaces with paralel mean curvature, Surfaces with constant mean curvature in , Local existence theorem for surfaces with constant mean curvature, axiom of spheresLocus of spheres, Canal hypersurfaces, Ricci curvature and scalar curvature for pseudoumbilical submanifold, Characterizations of umbilical submanifolds, The Gauss map, geometric inequalities, total mean curvature, Submanifolds with nonnegative scalar curvature,
 Dersin Öğrenme Çıktıları Öğretim Yöntemleri Ölçme Yöntemleri 1 - He/She synthesizes manifolds via differential geometry 1 - 2 - 3 - 8 - 15 - A - C - 2 - He/She defines operation on manifold. 1 - 2 - 3 - 8 - 15 - A - C - 3 - He/She computes the curvature of a manifold. 1 - 2 - 3 - 8 - 15 - A - C - 4 - He/she illustrates minimal submanifold 1 - 2 - 3 - 8 - 15 - A - C - 5 - He/She defines constant mean curvature in surface 1 - 2 - 8 - 15 - A - C - 6 - He/She defines sphere axiom 1 - 2 - 3 - 8 - 15 - A - C - 7 - He/She synthesizes surfaces via differential geometry 1 - 2 - 3 - 8 - 15 - A - C - 8 - He/she illustrates hypersurfaces 1 - 2 - 3 - 8 - 15 - A - C - 9 - He/She gives charecterization of a umbilicity submanifolds 1 - 2 - 3 - 8 - 15 - A - C - 10 - He/she illustrates stable hypersurfaces 1 - 2 - 3 - 8 - 15 - A - C -
 Öğretim Yöntemleri: 1:Lecture 2:Question-Answer 3:Discussion 8:Group Study 15:Problem Solving Ölçme Yöntemleri: A:Testing C:Homework

Ders Akışı

Hafta Konular ÖnHazırlık
1 Riemannian manifolds, Curvature tensor, Theorem of Frobenius
2 Induced connection and second fundamental form
3 Equations of Gauss, Codazzi and Ricci
4 Scalar curvature of submanifolds
5 Minimal submanifolds in Euclidean space, Minimal submanifolds of a submanifold
6 Surfaces with parallel mean curvature vector
7 Surfaces with constant mean curvature in R^3 , Local existence theorem for surfaces with constant mean curvature
8 Axiom of spheres, locus of the spheres
9 Canal hypersurfaces
10 Ricci curvature and scalar curvature for pseudoumbilical submanifolds
11 The average fixed curvature pseudoumbilical submanifolds
12 Characterizations of umbilical submanifolds
13 The Gauss map, Geometric inequalities, Total mean curvature
14 Submanifolds with nonnegative scalar curvature

Kaynaklar

Ders Notu 1. Şahin, B., Manifoldların Diferensiyel Geometrisi, Nobel Yayınları, Ekim 2012.
Ders Kaynakları 2. Chen, B., Geometry of Submanifolds, Marcel Dekker. Inc. New York, 1973.
3. Kobayashi, S., and Nomizu, K., Foundations of differential geometry, Number 15, Volume II, New York, 1969.
4. O’Neill B., Elementary Differential Geometry, Academic Press, New York, 1997.

Dersin Program Çıktılarına Katkısı

No Program Öğrenme Çıktıları KatkıDüzeyi
1 2 3 4 5
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. X
2 Student completes the missing or limited knowledge by using the scientific methods. X
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. X
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches. X
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. X
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters. X
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. X
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them. X
11 Student looks out for the scientific and ethic values while gathering, interpreting and publishing data. X

Değerlendirme Sistemi

YARIYIL İÇİ ÇALIŞMALARI SIRA KATKI YÜZDESİ
AraSinav 1 70
Odev 1 30
Toplam 100
Yıliçinin Başarıya Oranı 50
Finalin Başarıya Oranı 50
Toplam 100

AKTS - İş Yükü

Etkinlik Sayısı Süresi(Saat) Toplam İş yükü(Saat)
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 15 15
Assignment 1 10 10
Final examination 1 25 25
Toplam İş Yükü 146
Toplam İş Yükü /25(s) 5.84
Dersin AKTS Kredisi 5.84
; ;