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### Natural Almost Hermitian Structures on Conformally Foliated 4-Dimensional Lie Groups with Minimal Leaves

(2022) In Master's Theses in Mathematical Sciences MATM03 20212
Centre for Mathematical Sciences
Abstract
Let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation with minimal leaves. Let J be an almost Hermitian structure on G adapted to the foliation. The corresponding Lie algebra must then belong to one of 20 families according to S. Gudmundsson and M. Svensson. We classify such structures J which are almost Kähler, integrable or Kähler. Hereby, we construct 16 multi-dimensional almost Kähler families, 18 integrable families and 11 Kähler families.
Popular Abstract
In any scientific field, it is important to classify certain objects in order to form a greater understanding. Here, this is done for so-called 4-dimensional Lie groups with some additional specifications. The theory of Lie algebras and Lie groups is connected to almost all topics in mathematics and is also widely used in physics. For instance, gauge theory in physics uses certain symmetries of a physical system to form a Lie group. It is through the use of gauge theory that the existence of certain particles has been postulated, which points towards a greater understanding of our universe.

Throughout this work, we use a field in mathematics called Riemannian geometry. Euclidean geometry is what most people have been using when working... (More)
In any scientific field, it is important to classify certain objects in order to form a greater understanding. Here, this is done for so-called 4-dimensional Lie groups with some additional specifications. The theory of Lie algebras and Lie groups is connected to almost all topics in mathematics and is also widely used in physics. For instance, gauge theory in physics uses certain symmetries of a physical system to form a Lie group. It is through the use of gauge theory that the existence of certain particles has been postulated, which points towards a greater understanding of our universe.

Throughout this work, we use a field in mathematics called Riemannian geometry. Euclidean geometry is what most people have been using when working with
elementary geometry. Here, lines are assumed to be straight in the sense that two parallel lines will never meet, no matter how long you make them. It is, however, sometimes useful to work with geometry where lines can be assumed to not be straight. According to general relativity, for example, the path of a light particle bends when it is in the presence of a massive body. Such geometry is referred to as non-Euclidean geometry. Riemannian geometry was developed in the 19th century by Bernhard Riemann. In 1854, he presented his ideas in a lecture hall at his university in Göttingen. Among the audience was his former teacher Carl Friedrich Gauss. With his new theory, Riemann provided a system to unite all non-Euclidean geometries.

Riemannian geometry makes it possible to define a so-called manifold. A manifold is a space that up-close looks Euclidean. For instance, if a person stands on a sphere as big as the Earth and looks around, they appear to be standing on a flat plane. A Lie group is defined as a manifold that also has the properties of a group. A mathematical group is an arbitrary set paired with an operation (for example addition or multiplication) that is associative, has an identity element and any element must have an inverse. For example, the set of whole numbers paired with addition is a group. Any Lie group has an associated Lie algebra, which is the associated Euclidean space at a point of the Lie group. In the case of a person standing on a sphere of the size of the Earth, imagine that the person makes the perceived plane at the point at which they are standing infinitely larger.

The 4-dimensional Lie group studied in this thesis is also a so-called almost Hermitian manifold, which can be seen as a generalisation of a complex space. With the specifications that will be given to this Lie group, it has been shown by S. Gudmundsson and M. Svensson that its corresponding Lie algebras can be divided into 20 families. We use the work of A. Gray and L.M. Hervella to show that any 4-dimensional almost Hermitian manifold must belong to one of four classes and determine when each of the 20 families belongs to the four classes. (Less)
author
supervisor
organization
course
MATM03 20212
year
type
H2 - Master's Degree (Two Years)
subject
keywords
differential geometry, riemannian geometry, complex geometry, manifolds, lie groups, lie algebras, complex manifolds, complex structures, almost complex structures, almost complex manifolds, kähler manifolds, hermitian manifolds, almost hermitian structures, conformal foliations
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFMA-3128-2022
ISSN
1404-6342
other publication id
2022:E5
language
English
id
9075877
alternative location
https://www.matematik.lu.se/matematiklu/personal/sigma/students/Emma-Andersdotter-Svensson-MSc.pdf
2024-04-15 17:21:44
date last changed
2024-04-15 17:21:44
```@misc{9075877,
abstract     = {{Let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation with minimal leaves. Let J be an almost Hermitian structure on G adapted to the foliation. The corresponding Lie algebra must then belong to one of 20 families according to S. Gudmundsson and M. Svensson. We classify such structures J which are almost Kähler, integrable or Kähler. Hereby, we construct 16 multi-dimensional almost Kähler families, 18 integrable families and 11 Kähler families.}},
author       = {{Andersdotter Svensson, Emma}},
issn         = {{1404-6342}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Master's Theses in Mathematical Sciences}},
title        = {{Natural Almost Hermitian Structures on Conformally Foliated 4-Dimensional Lie Groups with Minimal Leaves}},
year         = {{2022}},
}

```