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Riemannian Geometry (ΓΕ3): Διαφορά μεταξύ των αναθεωρήσεων

Riemannian Geometry (ΓΕ3): Διαφορά μεταξύ των αναθεωρήσεων

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(Νέα σελίδα με 'Graduate Courses Outlines - [https://math.uoi.gr Department of Mathematics] === General === {| class="wikitable" |- ! School | School of Science |- ! Academic Unit | Department of Mathematics |- ! Level of Studies | Graduate |- ! Course Code | ΧΧΧ |- ! Semester | 000 |- ! Course Title | ΧΧΧ |- ! Independent Teaching Activities | Lectures (Weekly Teaching Hours: 3, Credits: 7.5) |- ! Course Type | General Background |- ! Prerequisite Courses | - |- ! L...')
 
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[[Graduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Γεωμετρία Riemann (ΓΕ3)|Ελληνική Έκδοση]]
{{Course-Graduate-Top-EN}}
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=== General ===
=== General ===
Γραμμή 15: Γραμμή 17:
|-
|-
! Course Code
! Course Code
| ΧΧΧ
| ΓΕ3
|-
|-
! Semester
! Semester
| 000
| 2
|-
|-
! Course Title
! Course Title
| ΧΧΧ
| Riemannian Geometry
|-
|-
! Independent Teaching Activities
! Independent Teaching Activities
Γραμμή 27: Γραμμή 29:
|-
|-
! Course Type
! Course Type
| General Background
| Special Background
|-
|-
! Prerequisite Courses
! Prerequisite Courses
Γραμμή 34: Γραμμή 36:
! Language of Instruction and Examinations
! Language of Instruction and Examinations
|
|
ΧΧΧ
Greek
|-
|-
! Is the Course Offered to Erasmus Students
! Is the Course Offered to Erasmus Students
| Yes
| Yes (in English)
|-
|-
! Course Website (URL)
! Course Website (URL)
Γραμμή 49: Γραμμή 51:
! Learning outcomes
! Learning outcomes
|
|
ΧΧΧ
In this lecture we introduce basic notions of Riemannian Geometry. More precisely, we introduce among others the notions of Riemannian metric, Levi-Civita connection, holonomy, curvature operator, Ricci curvature, sectional curvature, scalar curvature and Jacobi field.
|-
|-
! General Competences
! General Competences
|
|
ΧΧΧ
* Work autonomously
* Work in teams
* Develop critical thinking skills.
|}
|}


=== Syllabus ===
=== Syllabus ===


ΧΧΧ
* Riemannian metrics, isometries, conformal maps.
* Geodesics and exponential maps.
* Parallel transport and holonomy.
* Hopf-Rinow’s Theorem.
* Curvature operator, Ricci curvature, scalar curvature.
* Riemannian submanifolds.
* Gauss-Codazzi-Ricci equations.
* 1st and 2nd variation of length.
* Jacobi fields.
* Comparison theorems.
* Homeomorphic sphere theorem.


=== Teaching and Learning Methods - Evaluation ===
=== Teaching and Learning Methods - Evaluation ===
Γραμμή 66: Γραμμή 80:
! Delivery
! Delivery
|
|
ΧΧΧ
Face-to-face
|-
|-
! Use of Information and Communications Technology
! Use of Information and Communications Technology
|
| -
ΧΧΧ
|-
|-
! Teaching Methods
! Teaching Methods
Γραμμή 81: Γραμμή 94:
| 39
| 39
|-
|-
| ΧΧΧ
| Autonomous Study
| 000
| 78
|-
|-
| ΧΧΧ
| Solution of Exercises - Homeworks
| 000
| 70.5
|-
|-
| Course total  
| Course total  
Γραμμή 93: Γραμμή 106:
! Student Performance Evaluation
! Student Performance Evaluation
|
|
ΧΧΧ
Written final examination, presentations of HomeWorks.
|}
|}


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Τελευταία αναθεώρηση της 16:28, 15 Ιουνίου 2023

General

School School of Science
Academic Unit Department of Mathematics
Level of Studies Graduate
Course Code ΓΕ3
Semester 2
Course Title Riemannian Geometry
Independent Teaching Activities Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type Special Background
Prerequisite Courses -
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 Riemannian Geometry. More precisely, we introduce among others the notions of Riemannian metric, Levi-Civita connection, holonomy, curvature operator, Ricci curvature, sectional curvature, scalar curvature and Jacobi field.

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

Syllabus

  • Riemannian metrics, isometries, conformal maps.
  • Geodesics and exponential maps.
  • Parallel transport and holonomy.
  • Hopf-Rinow’s Theorem.
  • Curvature operator, Ricci curvature, scalar curvature.
  • Riemannian submanifolds.
  • Gauss-Codazzi-Ricci equations.
  • 1st and 2nd variation of length.
  • Jacobi fields.
  • Comparison theorems.
  • Homeomorphic sphere 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

Written final examination, presentations of HomeWorks.

Attached Bibliography

  • M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
  • J.-Η. Eschenburg, Comparison theorems in Riemannian Geometry, Lecture Notes, Universität Augsburg, 1994.
  • J. Jost, Riemannian Geometry and Geometric Analysis, Seventh edition, Universitext, Springer, 2017.
  • J. Lee, Riemannian manifolds: An introduction to curvature, Graduate Texts in Mathematics, 176, Springer, 1997.
  • P. Petersen, Riemannian Geometry, Third edition, Graduate Texts in Mathematics, 171, Springer, 2016.
  • Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.
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