ON EUCLIDEAN G-MANIFOLDS WHICH HAVE TWO DIMENSIONAL ORBIT SPACES

2011 ◽  
Vol 22 (03) ◽  
pp. 399-406
Author(s):  
R. MIRZAIE
Keyword(s):  
Euclidean Space ◽  
Half Plane ◽  
Lie Group ◽  
Orbit Space ◽  
Two Dimensional ◽  
Orbit Spaces ◽  
Group Of Isometries ◽  
Connected Lie Group

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

scholarly journals An introduction to spherical orbit spaces

2002 ◽  
Vol 32 (8) ◽  
pp. 453-469
Author(s):  
Jill Mcgowan ◽  
Catherine Searle
Keyword(s):  
Lie Group ◽  
Two Dimensional ◽  
Orbit Spaces ◽  
Quotient Spaces ◽  
Important Field ◽  
Connected Lie Group

Consider a compact, connected Lie groupGacting isometrically on a sphereSnof radius1. Two-dimensional quotient spaces of the typeSn/Ghave been investigated extensively. This paper provides an elementary introduction, for nonspecialists, to this important field by way of several classical examples and supplies an explicit list of all the isotropy subgroups involved in these examples.

scholarly journals Regularity of mean-values

1988 ◽  
Vol 45 (1) ◽  
pp. 117-126
Author(s):  
Christopher Meaney
Keyword(s):  
Euclidean Space ◽  
Mean Value ◽  
Dimensional Sphere ◽  
Mean Values ◽  
Regular Elements ◽  
Integrable Functions ◽  
Spherical Mean ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLetXbe either thed-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳf(x, a), wherex∈Xandavaries over a torusAin the group of isometries ofX. For each of these cases there is an intervalpO<p≦ 2, where thep0depends on the geometry ofX, such that iffis inLp(X) then there is a set full measure inXand ifxlies in this set, the function a ↦ℳf(x, a) has some Hölder continuity on compact subsets of the regular elements ofA.

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

scholarly journals Observability and Symmetries of Linear Control Systems

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 953
Author(s):  
Víctor Ayala ◽  
Heriberto Román-Flores ◽  
María Torreblanca Todco ◽  
Erika Zapana
Keyword(s):  
Euclidean Space ◽  
Control Systems ◽  
Lie Groups ◽  
Lie Group ◽  
Linear Control ◽  
Linear Control Systems ◽  
Euclidean Spaces ◽  
General Setup ◽  
Connected Lie Group

The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.

scholarly journals THE RADIAL PART OF THE ZERO-MODE HAMILTONIAN FOR SIGMA MODELS WITH GROUP TARGET SPACE

2004 ◽  
Vol 16 (05) ◽  
pp. 603-628 ◽  
Author(s):  
DOUG PICKRELL
Keyword(s):  
Sigma Models ◽  
Lie Group ◽  
Sigma Model ◽  
Zero Mode ◽  
Target Space ◽  
Two Dimensional ◽  
Radial Part ◽  
Group Target ◽  
Simply Connected ◽  
Connected Lie Group

In this note, we use geometric arguments to derive a possible form for the radial part of the "zero-mode Hamiltonian" for the two-dimensional sigma model with target space S3, or more generally a compact simply connected Lie group.

scholarly journals Almost free actions on manifolds

1974 ◽  
Vol 10 (2) ◽  
pp. 177-196 ◽  
Author(s):  
Philip T. Church ◽  
Klaus Lamotke
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Group ◽  
Orbit Space ◽  
Fixed Point Set ◽  
Point Set ◽  
Free Actions ◽  
Connected Lie Group

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and(2) the (even) number of fixed points.Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

Some unexpectedly non-singular orbit spaces

1977 ◽  
Vol 82 (1) ◽  
pp. 35-37
Author(s):  
H. R. Morton
Keyword(s):  
Lie Group ◽  
Orbit Space ◽  
Compact Lie Group ◽  
Orbit Spaces ◽  
Singular Orbit ◽  
Smooth Action ◽  
The Matrix ◽  
Orbit Types ◽  
Image Position

The orbit space of a manifold under the smooth action of a compact Lie group is typically a ‘manifold with singularities’. The orbits of a given type each form a mani-fold, but these ‘strata’ generally do not give a manifold when pieced together. For example, the orbit space of S3 under the action of given by the matrix.is the suspension of P(2), which fails to be a manifold at the suspension points.In this paper I shall give examples where, in spite of having many orbit types, the orbit space is a manifold, with or without boundary.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

Diffraction of a pulse by a resistive half-plane. I. Normal incidence

1960 ◽  
Vol 255 (1283) ◽  
pp. 538-549 ◽  
Keyword(s):  
Time Dependence ◽  
Half Plane ◽  
Wave Diffraction ◽  
Diffraction Problem ◽  
Normal Incidence ◽  
Step Function ◽  
Two Dimensional ◽  
Dynamic Similarity ◽  
Function Time ◽  
Dimensional Wave

The two-dimensional wave diffraction problem, acoustic or electromagnetic, in which a pulse of step-function time dependence is diffracted by a resistive half-plane is solved by assuming dynamic similarity in the solution.

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