Sumários
Holonomy - V
29 Maio 2024, 15:30 • Giordano Cotti
In this last lecture, we addressed the study of Riemannian holonomies. We formulated and discussed the remarkable "de Rham Decomposition Theorem". We concluded by presenting Berger's list of possible Riemannian holonomies for a simply-connected manifold that is irreducible and not locally a symmetric space. The lecture introduced the notions of Kähler, Calabi-Yau, hyper-Kähler, quaternion Kähler, and Einstein manifolds. Finally, we mentioned Dominic Joyce's recent works on the existence of G2 and Spin(7) manifolds.
Holonomy - IV
28 Maio 2024, 10:00 • Giordano Cotti
In this lecture, we focused on connections defined on the tangent bundle of a manifold. We introduced the notion of torsion, and proved the first and second Bianchi identities. Additionally, we introduced the concept of connections compatible with a metric, and formulated and proved the fundamental theorem regarding the existence and uniqueness of the Levi-Civita connection.
Holonomy - III
22 Maio 2024, 15:30 • Giordano Cotti
We continued and completed the proof of the Ambrose-Singer-Nijenhuis Theorem. As a deep result, we recalled Yamabe's theorem: an arcwise connected subgroup of a Lie group is a Lie subgroup.
Holonomy - II
21 Maio 2024, 10:00 • Giordano Cotti
In this lecture, we introduced the holonomy group of a connection at a point, as well as the restricted holonomy group. We formulated and began to prove the Ambrose-Singer-Nijenhuis Theorem: the restricted holonomy group is a connected Lie group with a Lie algebra determined by the curvature.
Holonomy - I
15 Maio 2024, 15:30 • Giordano Cotti
In this lecture, we addressed the following problem: given two homotopic curves and , what is the difference in the parallel transport along these curves? We showed that the curvature is responsible for this difference.