# Differential Geometry & Symmetry

#### Lecturers

Dr Romina Arroyo, The University of Queensland
Dr Ramiro Lafuente, The University of Queensland

#### Synopsis

Symmetries appear naturally in a wide range of mathematical and physical theories. They usually arise as groups acting as transformations of a certain space while preserving a given structure. Studying geometric spaces with symmetries not only provides us with a rich source of examples, but also gives rise to new rigidity phenomena and nicer structural properties which become interesting in their own right.

In this course we will cover the basic concepts in differential and Riemannian geometry. After that we will introduce Lie groups and homogeneous Riemannian manifolds (Riemannian manifolds with a transitive group of isometries), and describe their geometric properties. Our main focus will be in the study of state-of-the-art questions involving curvature and special metrics such as Einstein and Ricci solitons. Along the way, special attention will be given to important open problems in the field.

#### Course Overview

• Smooth manifolds, tangent vectors and spaces, vector fields
• Riemannian metrics, covariant derivative, curvature.
• Brief introduction to Lie groups, Lie algebras and homogeneous spaces
• The geometry of homogeneous Riemannian manifolds
• Homogeneous Einstein and Ricci soliton metrics

#### Prerequisites

• A basic course in differential geometry of curves and surfaces; linear algebra; multi-variable calculus. Some familiarity with Lie groups/algebras would help but is definitely not required.

#### Assessment

• Two assignments, 20% each.
• Final exam, 60%.

#### Resources/pre-reading (if available)

Lecture notes will be provided during the course.

• M. Do Carmo: Riemannian geometry
• J. Lee: Riemannian geometry
• Knapp, Lie groups beyond an introduction (Some sections in Chapter 1)
• Besse: Einstein manifolds (Chapter 7)

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Dr Romina Arroyo,
The University of Queensland

Romina M Arroyo is a CONICET Research Assistant and Assistant Professor at the National University of Cordoba, Argentina. Currently she is also a postdoc at the University of Queensland. She obtained her PhD in Cordoba in 2013, and held a visiting position at McMaster University, Canada in 2014-15. Her research focuses on studying geometric flows, distinguised metrics and Ricci curvature in homogeneous Riemannian manifolds.

Dr Ramiro Lafuente,
The University of Queensland

Ramiro is a Lecturer and an Australian Research Council DECRA fellow at the University of Queensland. He is originally from Argentina, where he obtained his PhD in 2013. He was an Alexander von Humboldt postdoctoral fellow in Münster, Germany between 2015 and 2017, and a research member at MSRI, Berkeley, CA in 2016. His main research interests are in Riemannian geometry with symmetries, but he also enjoys thinking about questions in a variety of topics including Lie theory, geometric analysis, complex differential geometry and geometric invariant theory.