SYMPLECTIC CONNECTIONS AND EXOTIC HOLONOMY

1998 ◽  
Vol 09 (01) ◽  
pp. 47-61 ◽  
Author(s):  
QUO-SHIN CHI
Keyword(s):  
Symplectic Connections ◽  
Torsion Free

We show that exotic holonomies arise naturally from torsion-free sympletic connections.

scholarly journals SYMPLECTIC CONNECTIONS

2006 ◽  
Vol 03 (03) ◽  
pp. 375-420 ◽  
Author(s):  
PIERRE BIELIAVSKY ◽  
MICHEL CAHEN ◽  
SIMONE GUTT ◽  
JOHN RAWNSLEY ◽  
LORENZ SCHWACHHÖFER
Keyword(s):  
Variational Principle ◽  
Critical Points ◽  
Ricci Tensor ◽  
Symmetric Spaces ◽  
Complex Geometry ◽  
Ricci Flat ◽  
Symplectic Connections

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far-reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to non-commutative symplectic symmetric spaces.

scholarly journals Remarks on Symplectic Connections

2006 ◽  
Vol 78 (3) ◽  
pp. 307-328 ◽  
Author(s):  
Simone Gutt
Keyword(s):  
Symplectic Connections

Special Symplectic Connections and Poisson Geometry

2004 ◽  
Vol 69 (1-3) ◽  
pp. 115-137 ◽  
Author(s):  
Michel Cahen ◽  
LORENZ J. SCHWACHH�FER
Keyword(s):  
Poisson Geometry ◽  
Symplectic Connections

scholarly journals Symplectic connections and the linearisation of Hamiltonian systems

1991 ◽  
Vol 117 (3-4) ◽  
pp. 329-380 ◽  
Author(s):  
J. E. Marsden ◽  
T. Ratiu ◽  
G. Raugel
Keyword(s):  
Hamiltonian System ◽  
Symplectic Structure ◽  
Equilibrium Solution ◽  
Hamiltonian Structure ◽  
Second Variation ◽  
First Variation ◽  
Space Construction ◽  
Symplectic Connections ◽  
The First Variation ◽  
The Second Variation

SynopsisThis paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

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