Riemannian manifolds with special holonomy

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Riemannian manifolds with special holonomy

Semester
SoSe 2026

Lecturer
Samuel Lockman

Type of course (Veranstaltungsart)
Vorlesung

German title
Riemannsche Mannigfaltigkeiten mit spezieller Holonomie

Contents
The holonomy group is the group given by parallel transport of vectors in a vector bundle along any loop. When the manifold carries a Riemannian metric and the vector bundle is the tangent bundle with the Levi-Civita connection, there is a remarkable and at first sight mysterious list containing the different possibilities for the holonomy group, called Berger's list. Each group on the list gives rise to a distinct geometric theory requiring different techniques to understand.
Beyond understanding geometry, special holonomy has been central in mathematical physics and is related to the reason why string theorists believe the universe to be either 10- or 11-dimensional. The original proofs of Berger's list are rather long and complicated, but in 2005, a shorter geometric proof was presented by Carlos Olmos. One goal of this course is to understand this proof. On the path to this result, we will:

develop the theory of connections and curvature (on vector and principal bundles),
explore the fundamental results in holonomy theory, such as the Ambrose-Singer theorem and the relationship between special holonomy groups and their corresponding parallel geometric structure,
understand symmetric spaces and their holonomy groups,

If time permits, further topics will be explored depending on the interests of the audience.

Literature

Carlos Olmos, A geometric proof of the Berger Holonomy Theorem, Ann. Math. 161 (2005), 579-588, https://doi.org/10.4007/annals.2005.161.579
Dominique Joyce, Compact Manifolds with Special Holonomy
Berndt, Console, and Olmos, Submanifolds and Holonomy

More literature will be given during the course.

Recommended previous knowledge
Analysis I, II and IV Linear Algebra I and II Differential Geometry I

Time/Date
Tuesday, 16-18

Location
M102

Additional question session
Time/Date: Wednesday 14-16
Location: M009

Registration

Registration for course work/examination/ECTS: FlexNow

Course work (Studienleistungen)

Successful participation in the exercise classes: at least 50% of the fortnightly exercises

Examination (Prüfungsleistungen)

Oral exam: Duration: 30 minutes, Date: individual arrangements, re-exam: Date:

Modules
MV, MGAGeo

ECTS
4,5 ECTS

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