Elementary Differential Geometry (MAY522)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAY522
Semester	5
Course Title	Elementary differential geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	http://users.uoi.gr/tvlachos/

Learning Outcomes

Learning outcomes

It is an introductory course on differential geometry. The aim is to introduce and study geometric properties of regular curves (both plane and space) and regular surfaces. Fundamental notions of differential geometry of curves and surfaces are introduced and studied. Among them is the notion of curvature. The study requires tools from Linear Algebra and Calculus of several variables. Upon completion of the course, the student should be familiar with basic notions of differential geometry like the one of curvature, first and second fundamental form, isometries between surfaces and their geometric meaning.

General Competences

Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Plane curves, arclength, curvature, Frenet frame.
• Space curves, curvature and torsion, Frenet frame, fundamental theorem of curves.
• Surfaces, parametrization, Gauss map, Weingarten map, first and second fundamental form, normal curvature, principal and asymptotic directions, Gaussian and mean curvature, minimal surfaces, Theorema Egregium, Gauss and Weingarten formulas, fundamental theorem of surfaces, developable surfaces.

Teaching and Learning Methods - Evaluation

Delivery	Direct
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 65 Autonomous study 127.5 Course total 187.5
Student Performance Evaluation	Written final examination

Attached Bibliography

• Δ. Κουτρουφιώτης, Στοιχειώδης Διαφορική Γεωμετρία, Εκδόσεις Leader Books, 2006
• Barrett O' Neil, Στοιχειώδης Διαφορική Γεωμετρία, Πανεπιστημιακές Εκδόσεις Κρήτης, 2002
• Andrew Pressley, Στοιχειώδης Διαφορική Γεωμετρία, Πανεπιστημιακές Εκδόσεις Κρήτης, 2011
• Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976