Differentiable Manifolds (MAE728)

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School of Science

Academic Unit

Department of Mathematics

Level of Studies


Course Code




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


Course Website (URL) See eCourse, the Learning Management System maintained by the University of Ioannina.

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.


  • 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


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 the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

  • 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.