Canonical reduction on cotangent symplectic manifolds with group action and on associated principal fiber bundles with connection

2006 ◽  
Vol 9 (1) ◽  
pp. 96-106 ◽  
Author(s):  
Ya. A. Prykarpats’kyi
Keyword(s):  
Group Action ◽  
Symplectic Manifolds ◽  
Fiber Bundles ◽  
Principal Fiber Bundles

CLASSIFICATION OF GAUGE-RELATED INVARIANT CONNECTIONS

1993 ◽  
Vol 05 (01) ◽  
pp. 69-103 ◽  
Author(s):  
R. BAUTISTA ◽  
J. MUCIÑO ◽  
E. NAHMAD-ACHAR ◽  
M. ROSENBAUM
Keyword(s):  
Symmetry Group ◽  
Moduli Space ◽  
Structure Group ◽  
Fiber Bundles ◽  
Explicit Solutions ◽  
Arbitrary Structure ◽  
Holonomy Groups ◽  
Principal Fiber Bundles

Connection 1-forms on principal fiber bundles with arbitrary structure groups are considered, and a characterization of gauge-equivalent connections in terms of their associated holonomy groups is given. These results are then applied to invariant connections in the case where the symmetry group acts transitively on fibers, and both local and global conditions are derived which lead to an algebraic procedure for classifying orbits in the moduli space of these connections. As an application of the developed techniques, explicit solutions for SU (2) × SU (2)-symmetric connections over S2 × S2, with SU(2) structure group, are derived and classified into non-gauge-related families, and multi-instanton solutions are identified.

scholarly journals The Guillemin–Sternberg conjecture for noncompact groups and spaces

2008 ◽  
Vol 1 (3) ◽  
pp. 473-533 ◽  
Author(s):  
P. Hochs ◽  
N.P. Landsman
Keyword(s):  
Normal Subgroup ◽  
Dirac Operator ◽  
Lie Groups ◽  
Group Action ◽  
Symplectic Manifolds ◽  
Lie Group Action ◽  
Compact Case ◽  
Assembly Map ◽  
High Degree ◽  
Special Case

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

Variational principles on principal fiber bundles

1987 ◽  
Vol 4 (2) ◽  
pp. 183-205 ◽  
Author(s):  
Hernan Cendra ◽  
Alberto Ibort ◽  
Jerrold Marsden
Keyword(s):  
Variational Principles ◽  
Fiber Bundles ◽  
Principal Fiber Bundles

scholarly journals Infinitesimal Holonomy Groups of Bundle Connections

1956 ◽  
Vol 10 ◽  
pp. 105-123 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Differential Geometry ◽  
Present Note ◽  
Holonomy Group ◽  
Fiber Bundles ◽  
Linear Connections ◽  
Holonomy Groups ◽  
Principal Fiber Bundles ◽  
Sharpened Form

In Introduction In differential geometry of linear connections, A. Nijenhuis has introduced the concepts of local holonomy group and infinitesimal holonomy group and obtained many interesting results [6].The purpose of the present note is to generalize his results to the case of connections in arbitrary principal fiber bundles with Lie structure groups. The concept of local holonomy group can be immediately generalized and has been already utilized by S. Kobayashi [4]. Our main results are Theorems 4 and 5 on infinitesimal holonomy groups. The proofs depend on a little sharpened form of a theorem of Ambrose-Singer [1]. In the case of linear connections, our infinitesimal holonomy group coincides with that of Nijenhuis, as we shall show in Section 6.

THE YANG–MILLS FUNCTIONAL AND BOGOMOLOV INEQUALITY FOR ARBITRARY PRINCIPAL BUNDLES OVER KÄHLER MANIFOLDS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360015
Author(s):  
CARLOS TEJERO PRIETO
Keyword(s):  
Type Inequality ◽  
Structure Group ◽  
Fiber Bundles ◽  
Characteristic Classes ◽  
Compact Lie Group ◽  
Absolute Minimum ◽  
Yang Mills ◽  
Group A ◽  
Absolute Minimizers ◽  
Principal Fiber Bundles

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal Kℂ-bundle has to satisfy this generalized Bogomolov type inequality.

Curvature and torsion pseudotensors of coaffine connection in tangent bundle of hypercentred planes manifold

2020 ◽  
pp. 49-57
Author(s):  
A. V. Vyalova
Keyword(s):  
Differential Equations ◽  
Projective Space ◽  
Linear Group ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Smooth Manifold ◽  
Principal Bundle ◽  
Fiber Bundles ◽  
Curvature And Torsion ◽  
Principal Fiber Bundles

The hypercentered planes family, whose dimension coincides with dimension of generating plane, is considered in the projective space. Two principal fiber bundles arise over it. Typical fiber for one of them is the stationarity subgroup for hypercentered plane, for other — the linear group operating in each tangent space to the manifold. The latter bundle is called the principal bundle of linear coframes. The structural forms of two bundles are related by equations. It is proved that hypercentered planes family is a holonomic smooth manifold. In the principal bundle of linear coframes the coaffine connection is given. From the differential equations it follows that the coaffine connec­tion object forms quasipseudotensor. It is proved that the curvature and torsion objects for the coaffine connection in the linear coframes bundle form pseudotensors.

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