MATHEMATICAL EXCHANGE FORUM HOLONOMY OF RIEMANNIAN MANIFOLD
Date: October 30, 2026 Time: 12:00 PM Location: https://lums-edu-pk.zoom.us/j/91563808856?pwd=1lqShAhxUXqUTgG8NWaWPwpc0RkbKi.1
Speaker: Rashid Talha
How much of the geometry of a Riemannian manifold is encoded by parallel transport around loops?
The notion of holonomy addresses this question and reveals deep connections between curvature
and the geometry of the underlying manifold.
In this talk, we will introduce holonomy in the setting of Riemannian manifolds equipped with
the Levi–Civita connection. We will begin by studying parallel transport of tensors, and then
discuss the holonomy principle. The emphasis will be on examples and applications rather than
detailed proofs. After briefly introducing the general theory of holonomy, we will present Berger’s
classification of Riemannian holonomy groups and discuss the significance of its different classes
from the perspectives of mathematics and physics.
The aim is to introduce the audience to this beautiful area of mathematics, which originates
in differential geometry but is deeply connected with representation theory, Lie theory, PDEs,
differential topology, and physics, and to encourage them to explore these connections further.
Familiarity with smooth manifolds, Riemannian metrics, the Levi–Civita connection, and introduc-
tory group theory will be helpful.
30
Oct
Date: October 30, 2026
Time: 12:00 PM
Location: https://lums-edu-pk.zoom.us/j/91563808856?pwd=1lqShAhxUXqUTgG8NWaWPwpc0RkbKi.1
Speaker: Rashid Talha