Differentiable Manifolds (MAE728)

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Undergraduate Courses Outlines - Department of Mathematics

General

School

School of Science

Academic Unit

Department of Mathematics

Level of Studies

Undergraduate

Course Code

MAE728

Semester

7

Course Title

Differentiable Manifolds

Independent Teaching Activities

Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6)

Course Type

Special Background

Prerequisite Courses -
Language of Instruction and Examinations

Greek, English

Is the Course Offered to Erasmus Students

Yes

Course Website (URL) http://users.uoi.gr/ansavas/lectures/id-5.html

Learning Outcomes

Learning outcomes

In this lecture we introduce basic notions of modern Differential Geometry. More precisely, we introduce among others the notions of manifold, tangent bundle, connection, parallel transport and Riemannian metric.

General Competences
  • work autonomously
  • work in teams
  • develop critical thinking skills.

Syllabus

  • Smooth manifolds.
  • Smooth maps.
  • Tangent vectors.
  • Vector fields.
  • Regular values and Sard's Theorem.
  • Homotopy and Isotopy.
  • Lie bracket.
  • Frobenius' Theorem.
  • Connections and parallel transport.
  • Riemannian metrics.

Teaching and Learning Methods - Evaluation

Delivery

Classroom (face-to-face)

Use of Information and Communications Technology -
Teaching Methods
Activity Semester Workload
Lectures 39
Autonomous Study 111
Course total 150
Student Performance Evaluation

Weakly homeworks and written final examination.

Attached Bibliography

See Eudoxus. Additionally:

  • M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
  • V. Guillemin & A. Pollack, Differentiable Topology, Prentice-Hall, Inc, Englewood Cliffs, 1974.
  • J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, 2013.
  • J. Milnor, Topology From the Differentiable Viewpoint, Princeton University Press, NJ, 1997.
  • L. Tu, An Introduction to Manifolds, Universitext. Springer, New York, 2011.
  • Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.