Riemannian geometry, spring 2013

Lecturer

Ilkka Holopainen

Scope

10 cu.

Type

Advanced studies

Prerequisites

Vector analysis, topology, linear algebra, some knowledge on smooth manifolds
 (like the course "Johdatus differentiaaligeometriaan") and on differential equations.

Lectures

Weeks 3-9 and 11-18, Wed 10-12, Fri 12-14 in room C124.

Easter holiday 28.3.-3.4.

Exams

The course can be passed by an exam.

Content

• Differentiable manifolds (brief review)
• Riemannian metrics
• Connections
• Geodesics
• Curvature
• Jacobi fields
• Curvature and topology
• Comparison geometry
• The sphere theorem

Bibliography

• DoCarmo: Riemannian geometry, Birkhäuser, 1992
• Lee: Riemannian manifolds, An Introduction to Curvature, Springer, 1997

Lecture notes (first 101 pages): RiemannGeo.pdf

Registration

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Exercise groups

Home work assignments and solutions

Exercise 1  

Exercise 2  

Exercise 3  

Exercise 4  

Exercise 5   

Exercise 6   

Exercise 7   

Exercise 8   

Exercise 9   

Exercise 10  

Exercise 11  

Exercise 12