Classical Differential Geometry II

Course Information
TitleΚΛΑΣΙΚΗ ΔΙΑΦΟΡΙΚΗ ΓΕΩΜΕΤΡΙΑ ΙΙ / Classical Differential Geometry II
Cycle / Level1st / Undergraduate
Teaching PeriodSpring
Course ID40000470

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

Registered students: 15
OrientationAttendance TypeSemesterYearECTS
CoreElective Courses belonging to the selected specializationSpring-5.5

Class Information
Academic Year2020 – 2021
Class PeriodSpring
Faculty Instructors
Weekly Hours3
Class ID
Type of the Course
  • Scientific Area
Course Category
Specific Foundation / Core
Mode of Delivery
  • Face to face
  • Distance learning
The course is also offered to exchange programme students.
Language of Instruction
  • Greek (Instruction, Examination)
  • English (Instruction, Examination)
  • French (Instruction, Examination)
  • German (Instruction, Examination)
Required Courses
  • 0108 Linear Algebra
  • 0303Α Classical Differential Geometry I
Learning Outcomes
Upon successful completion of the course, students will be able to: 1) Use the notion of covariant derivative between vector fields and understand its functionality on a surface. 2) Compute and identify geodesic curves of a surface. They will also be able to ompute the geodesic curvature. 3) Apply the Gauss - Bonnet Theorem. 4) Distinguish between the spaces of constant curvature and their individual features. 5) Demonstrate these features on the basic models of such spaces.
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)
Reminder of basic constructions of Differential Geometry (principal and mean curvature, Gauss curvature, normal and geodesic curvature), covariant derivative and geodesic curves, length functional, Clairaut's Theorem, local and global Gauss-Bonnet Theorem, surfaces of constant curvature, topological structure of surfaces, Euler characteristic.
Curvature, Gauss-Bonnet Theorem, Euler characteristic.
Educational Material Types
  • Notes
  • Book
Use of Information and Communication Technologies
Use of ICT
  • Use of ICT in Communication with Students
Course Organization
Reading Assigment150.5
Student Assessment
Written examination
Student Assessment methods
  • Written Assignment (Summative)
  • Written Exam with Problem Solving (Summative)
Course Bibliography (Eudoxus)
- A. Pressley: Στοιχειώδης Διαφορική Γεωμετρία.Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2011 - B. O'Neill: Στοιχειώδης Διαφορική Γεωμετρία, Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2002 - Α. Αρβανιτογεώργος: Στοιχειώδης Διαφορική Γεωμετρία, Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, 2015 - Σ. Σταματάκης: Εισαγωγή στην Κλασική Διαφορική Γεωμετρία, Θεσσαλονίκη, Εκδόσεις Αϊβάζη, 2008 - Δ. Κουτρουφιώτης: Στοιχειώδης διαφορική γεωμετρία, Αθήνα : Leader Books, 2006
Additional bibliography for study
- M. Abate, F. Tovena: Curves and Surfaces. Springer, 2012 - M. P. do Carmo: Differential Geometry of Curves and Surfaces. Prentice – Hall, 1976 - J. Oprea: Differential Geometry and its Applications. Prentice Hall, 1997
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