Classical Differential Geometry I

Course Information
TitleΚΛΑΣΙΚΗ ΔΙΑΦΟΡΙΚΗ ΓΕΩΜΕΤΡΙΑ Ι / Classical Differential Geometry I
Cycle / Level1st / Undergraduate
Teaching PeriodWinter/Spring
Course ID600019647

Programme of Study: UPS of School of Mathematics (2014-today)

Registered students: 601
OrientationAttendance TypeSemesterYearECTS
CoreCompulsory Course536.5

Class Information
Academic Year2020 – 2021
Class PeriodWinter
Faculty Instructors
Weekly Hours5
Class ID
1. ΤΜΗΜΑ ΑEfthimios Kappos
2. ΤΜΗΜΑ ΒPanagiotis Batakidis
Course Type 2016-2020
  • Scientific Area
Course Type 2011-2015
Specific Foundation / Core
Mode of Delivery
  • Face to face
The course is also offered to exchange programme students.
Language of Instruction
  • Greek (Instruction, Examination)
  • English (Instruction, Examination)
Required Courses
  • 0301 Analytic Geometry I
  • 0302 Analytic Geometry II
Learning Outcomes
Upon successful completion of the course, students will: 1) Be able to recognize parametrizations of standard curves and surfaces. 2) Be able to check whether a curve is given with its natural parameter and if not, how to reparametrize it so. 3) Be able to calculate the basic geometric data, like the curvature, of a plane curve. 4) Be able to calculate the basic geometric data, like the curvature and torsion of a space curve. 5) Be able to classify space curves and manipulate the Frenet frame of a space curve. 6) Be able to check whether a parametrization corresponds to a smooth surface and compute its tangent plane. 7) Be able to compute distance between two points on a surface, length of a curve, and angle of two surface curves. 8) Be able to manipulate the basic geometric data of a surface like the normal vector and orientation, the Gauss map and shape operator, the first and second fundamental form, Gauss curvature, and mean curvature. 9) Be able to check which analytical data correspond to smooth surfaces (and in which way).
General Competences
  • Apply knowledge in practice
  • Make decisions
  • Work autonomously
  • Work in teams
  • Work in an international context
  • Work in an interdisciplinary team
  • Generate new research ideas
  • Advance free, creative and causative thinking
Course Content (Syllabus)
Theory of Curves: The concept of the curve in the differential geometry. The moving frame. The Frenet formulae. The fundamental theorem (existence and uniqueness). Osculating cycle. Plane curves. Theory of surfaces: The concept of surface in differential geometry. Curves on a surface. The first and the second fundamental form. Gauss, mean curvature and principal curvatures. Christoffel symbols. The Gauss map and equations of Gauss and Weingarten. Theorema Egregium of Gauss. The fundamental theorem (existence and uniqueness).
Theory of curves, theory of surfaces
Educational Material Types
  • Book
Use of Information and Communication Technologies
Use of ICT
  • Use of ICT in Communication with Students
Course Organization
Reading Assigment1153.8
Student Assessment
Written examination
Student Assessment methods
  • Written Exam with Multiple Choice Questions (Summative)
  • Written Exam with Short Answer Questions (Summative)
  • Written Exam with Extended Answer Questions (Summative)
  • Written Assignment (Summative)
  • Written Exam with Problem Solving (Summative)
Course Bibliography (Eudoxus)
- Σ. Σταματάκη: Εισαγωγή στην Κλασική Διαφορική Γεωμετρία, Θεσσαλονίκη, Εκδόσεις Αϊβάζη, 2008 - Ν. Στεφανίδη: Διαφορική Γεωμετρία, Β’ έκδοση βελτ. και επαυξ. Θεσσαλονίκη, 2014 - A. Pressley: Στοιχειώδης Διαφορική Γεωμετρία.Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2011 - B. O'Neill: Στοιχειώδης Διαφορική Γεωμετρία, Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2002
Additional bibliography for study
- M. P. do Carmo: Differential Geometry of Curves and Surfaces. Prentice – Hall, 1976 - J. Oprea: Differential Geometry and its Applications. Prentice Hall, 1997 - Β. Παπαντωνίου: Διαφορική Γεωμετρία, Πάτρα : Εκδόσεις Πανεπιστημίου Πατρών, 1996- 1997 - G. Στάμου: Ασκήσεις Διαφορικής Γεωμετρίας. Εκδόσεις Ζήτη, 1990
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