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Differential Geometry (ΓΕ2): Διαφορά μεταξύ των αναθεωρήσεων

Differential Geometry (ΓΕ2): Διαφορά μεταξύ των αναθεωρήσεων

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[[Graduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Διαφορική Γεωμετρία (ΓΕ2)|Ελληνική Έκδοση]]
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=== General ===
=== General ===
Γραμμή 30: Γραμμή 32:
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! Prerequisite Courses
! Prerequisite Courses
| Linear Algebra, Topology, Elementary differential geometry, Calculus, Analysis of several variables.
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Linear Algebra, Topology, Calculus of Several Variables.
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! Language of Instruction and Examinations
! Language of Instruction and Examinations
Γραμμή 47: Γραμμή 50:
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! Learning outcomes
! Learning outcomes
| This course introduces the basic notions of differential and Riemannian geometry.
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In this lecture we introduce basic notions of Differential Geometry. More precisely, we introduce among others the notions of manifold, manifold with boundary, vector bundle, connection, parallel transport, submanifold, differential form and de Rham cohomology.
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! General Competences
! General Competences
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# Work autonomously
* Work autonomously
# Work in teams
* Work in teams
# Develop critical thinking skills.
* Develop critical thinking skills.
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=== Syllabus ===
=== Syllabus ===


Differentiable manifolds, immersions, embeddings, submanifolds, vector fields, orientation covering spaces, partition of unity, Riemannian manifolds, Levi-Civita connection, curvature tensor, geodesics, exponential map, Isometric immersions, second fundamental form, hypersurfaces, Gauss, Codazzi and Ricci equations, applications.
* Topological and smooth manifolds.
* Tangent and cotangent bundles.
* Vector fields and their flows.
* Submanifolds and Frobenius’ Theorem.
* Vector bundles.
* Connection and parallel transport.
* Differential forms.
* De Rham cohomology.
* Integration.
* Stokes’ Theorem.


=== Teaching and Learning Methods - Evaluation ===
=== Teaching and Learning Methods - Evaluation ===
Γραμμή 65: Γραμμή 78:
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! Delivery
! Delivery
| Direct
| Face-to-face.
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! Use of Information and Communications Technology
! Use of Information and Communications Technology
Γραμμή 79: Γραμμή 92:
| 39
| 39
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| Study and analysis of bibliography
| Autonomous Study
| 78
| 78
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| Preparation of assignments and interactive teaching
| Solution of Exercises - Homeworks
| 70.5
| 70.5
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Γραμμή 90: Γραμμή 103:
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! Student Performance Evaluation
! Student Performance Evaluation
| Written final examination.
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Weakly HomeWorks, presentation in the blackboard of the HomeWorks, written final examination.
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Τελευταία αναθεώρηση της 16:29, 15 Ιουνίου 2023

General

School School of Science
Academic Unit Department of Mathematics
Level of Studies Graduate
Course Code ΓΕ2
Semester 1
Course Title Differential Geometry
Independent Teaching Activities Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type Special Background
Prerequisite Courses

Linear Algebra, Topology, Calculus of Several Variables.

Language of Instruction and Examinations Greek
Is the Course Offered to Erasmus Students Yes (in English)
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 Differential Geometry. More precisely, we introduce among others the notions of manifold, manifold with boundary, vector bundle, connection, parallel transport, submanifold, differential form and de Rham cohomology.

General Competences
  • Work autonomously
  • Work in teams
  • Develop critical thinking skills.

Syllabus

  • Topological and smooth manifolds.
  • Tangent and cotangent bundles.
  • Vector fields and their flows.
  • Submanifolds and Frobenius’ Theorem.
  • Vector bundles.
  • Connection and parallel transport.
  • Differential forms.
  • De Rham cohomology.
  • Integration.
  • Stokes’ Theorem.

Teaching and Learning Methods - Evaluation

Delivery Face-to-face.
Use of Information and Communications Technology -
Teaching Methods
Activity Semester Workload
Lectures 39
Autonomous Study 78
Solution of Exercises - Homeworks 70.5
Course total 187.5
Student Performance Evaluation

Weakly HomeWorks, presentation in the blackboard of the HomeWorks, written final examination.

Attached Bibliography

  • M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
  • J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
  • J. Lee, Introduction to smooth manifolds, Second edition, Graduate Texts in Mathematics, 218, Springer, 2013.
  • Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.
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