Invariant structures on Lie groups

In this thesis we consider nonholonomic Riemannian manifolds, and in particular, left- invariant nonholonomic Riemannian structures on Lie groups. These structures are closely related to mechanical systems with (positive definite) quadratic Lagrangians and nonholo- nomic constraints linear in velocities. In the first chapter, we review basic concepts of non- holonomic Riemannian geometry, including the left-invariant structures. We also examine the class of left-invariant structures with so-called Cartan-Schouten connections. The second chapter investigates the curvature of nonholonomic Riemannian manifolds and the Schouten and Wagner curvature tensors. The Schouten tensor is canonically associated to every non- holonomic Riemannian structure (in particular, we use it to define isometric invariants for structures on three-dimensional manifolds). By contrast, the Wagner tensor is not generally intrinsic, but can be used to characterise flat structures (i.e., those whose associated parallel transport is path-independent). The third chapter considers equivalence of nonholonomic Rie- mannian manifolds, particularly up to nonholonomic isometry. We also introduce the notion of a nonholonomic Riemannian submanifold, and investigate the conditions under which such a submanifold inherits its geometry from the enveloping space. The latter problem involves the concept of a geodesically invariant distribution, and we show it is also related to the curvature. In the last chapter we specialise to three-dimensional nonholonomic Riemannian manifolds. We consider the equivalence of such structures up to nonholonomic isometry and rescaling, and classify the left-invariant structures on the (three-dimensional) simply connected Lie groups. We also characterise the flat structures in three dimensions, and then classify the flat structures on the simply connected Lie groups. Lastly, we consider three typical examples of (left-invariant) nonholonomic Riemannian structures on three-dimensional Lie groups, two of which arise from problems in classical mechanics (viz., the Chaplygin problem and the Suslov problem).

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Lie Groups, Lie Algebras, Cohomology and some Applications in Physics

10.1017/cbo9780511599897 ◽

1995 ◽

Cited By ~ 90

Author(s):

Josi A. de Azcárraga ◽

Josi M. Izquierdo

Keyword(s):

Lie Groups ◽

Lie Algebras

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Disequilibrium and Divinity Salience as Invariant Structures in The Halal Executives’ Experience of Eudaimonia

ADVANCES IN BUSINESS RESEARCH INTERNATIONAL JOURNAL ◽

10.24191/abrij.v5i1.9995 ◽

2019 ◽

Vol 5 (1) ◽

pp. 109

Author(s):

Jalilah Ahmad ◽

Rosmimah Mohd. Roslin ◽

Mohd Ali Bahari Abdul Kadir

Keyword(s):

Data Analysis ◽

Lived Experience ◽

Status Quo ◽

The Self ◽

Phenomenological Method ◽

Sustainable Practices ◽

Muslim Population ◽

Living Well ◽

Hard Skills ◽

Invariant Structures

The global Halal industry is large and continues to grow as the global Muslim population increases in size and dispersion. There are 1.84 billion Muslims today spread over 200 countries and is expected to increase to 2.2 billion by 2030. The industry will be worth USD6.4 trillion by the end of 2018 with more non-traditional players and emergent markets. The stakes are high with pressures to generate novel and sustainable practices. This goes beyond systems and hard skills as it needs to cut into the self – the person of virtues in virtuous acts, not because they “have to” but because it is the purpose of humankind or his telos - to be “living well” and “acting well” or eudaimonia. This study seek to explore Halal executives’ lived experience of “eudaimonia.”. Using Giorgi’s descriptive psychological phenomenological method for data analysis, the study elicits two distinct invariant structures – ‘disequilibrium in status quo’ and ‘divinity salience’.

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