Contributions to the study of nonholonomic Riemannian manifolds

2017 ◽  
Author(s):  
◽  
Dennis Ian Barrett
Keyword(s):  
Lie Groups ◽  
Riemannian Manifolds ◽  
Parallel Transport ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Suslov Problem ◽  
Flat Structures ◽  
Invariant Structures ◽  
Simply Connected

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).

scholarly journals r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

scholarly journals LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 88-95
Author(s):  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Lie Groups ◽  
Complex Structure ◽  
New Left ◽  
Riemannian Metrics ◽  
Nilpotent Lie Groups ◽  
Complex Structures ◽  
Structural Group ◽  
Almost Complex ◽  
Flat Structures ◽  
Simply Connected

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

scholarly journals ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

2011 ◽  
Vol 08 (03) ◽  
pp. 501-510 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Three Dimensional ◽  
Flag Curvature ◽  
Nilpotent Lie Groups ◽  
Randers Metric ◽  
Levi Civita Connection ◽  
Simply Connected ◽  
Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

scholarly journals Left invariant Randers metrics of Berwald type on tangent Lie groups

2017 ◽  
Vol 15 (01) ◽  
pp. 1850015
Author(s):  
Farhad Asgari ◽  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Randers Metric ◽  
Tangent Lie Group ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected ◽  
Randers Metrics ◽  
Tangent Bundles

Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

scholarly journals Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

2021 ◽  
Vol 932 ◽  
Author(s):  
Idan S. Wallerstein ◽  
Uri Keshet
Keyword(s):  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Four Dimensions ◽  
Potential Flows ◽  
Prolate Spheroids ◽  
Flow Potential ◽  
Inverse Radius ◽  
Rayleigh Expansion ◽  
Simply Connected

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups

2019 ◽  
Vol 16 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Lie Groups ◽  
Semidirect Product ◽  
Three Dimensional ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
The Matrix ◽  
Canonical Foliation ◽  
Matrix Formula ◽  
Simply Connected

The main result of this paper gives a characterization of left-invariant almost [Formula: see text]-coKähler structures on three-dimensional (3D) semidirect product Lie groups [Formula: see text] in terms of the matrix [Formula: see text]. Then, we study the harmonicity of the Reeb vector field [Formula: see text] of a simply connected homogeneous almost [Formula: see text]-coKähler three-manifold, in terms of the Gaussian curvature of the canonical foliation.

scholarly journals Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Three Dimensional ◽  
Isotropy Subgroup ◽  
Left Translation ◽  
Isometry Groups ◽  
Group Automorphism ◽  
Group Of Automorphisms ◽  
Simply Connected

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

scholarly journals TOPOLOGICAL LOOPS HAVING SOLVABLE INDECOMPOSABLE LIE GROUPS AS THEIR MULTIPLICATION GROUPS

2020 ◽  
Author(s):  
Á. FIGULA ◽  
A. AL-ABAYECHI
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Abelian Extension ◽  
Solvable Lie Groups ◽  
Multiplication Group ◽  
Simply Connected

Abstract We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.

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