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 Variational principles and symmetries on fibered multisymplectic manifolds

2016 ◽  
Vol 24 (2) ◽  
pp. 137-152 ◽  
Author(s):  
Jordi Gaset ◽  
Pedro D. Prieto-Martínez ◽  
Narciso Román-Roy
Keyword(s):  
Variational Principles ◽  
Variational Calculus ◽  
Field Theories ◽  
Fiber Bundles ◽  
Conserved Quantities ◽  
Noether Symmetries ◽  
Geometric Framework ◽  
Gauge Symmetries ◽  
Special Cases ◽  
Standard Techniques

Abstract The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.

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 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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