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 Ekeland's variational principle and critical points of dynamical systems in locally complete spaces

2015 ◽  
Vol 6 (4) ◽  
pp. 107-113
Author(s):  
C. Bosch ◽  
C.L. García ◽  
F. Garibay-Bonales ◽  
C. Gómez-Wulschner ◽  
R. Vera
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Critical Points ◽  
Ekeland’S Variational Principle ◽  
Ekeland's Variational Principle

scholarly journals CHARACTERIZATIONS OF RICCI FLAT METRICS AND LAGRANGIAN SUBMANIFOLDS IN TERMS OF THE VARIATIONAL PROBLEM

2014 ◽  
Vol 57 (3) ◽  
pp. 643-651
Author(s):  
TETSUYA TANIGUCHI ◽  
SEIICHI UDAGAWA
Keyword(s):  
Variational Problem ◽  
Critical Points ◽  
Einstein Metrics ◽  
Lagrangian Submanifolds ◽  
Volume Element ◽  
Ricci Flat

AbstractGiven the pair (P, η) of (0,2) tensors, where η defines a volume element, we consider a new variational problem varying η only. We then have Einstein metrics and slant immersions as critical points of the 1st variation. We may characterize Ricci flat metrics and Lagrangian submanifolds as stable critical points of our variational problem.

A second derivative formula of the Liouville entropy at spaces of constant negative curvature

1997 ◽  
Vol 17 (5) ◽  
pp. 1131-1135 ◽  
Author(s):  
GERHARD KNIEPER
Keyword(s):  
Critical Points ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Symmetric Spaces ◽  
Local Maximum ◽  
Second Derivative ◽  
Constant Negative Curvature ◽  
Locally Symmetric Spaces ◽  
Derivative Formula ◽  
Total Scalar Curvature

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

scholarly journals Minimax theorems onC1manifolds via Ekeland variational principle

2003 ◽  
Vol 2003 (13) ◽  
pp. 757-768 ◽  
Author(s):  
Mabel Cuesta
Keyword(s):  
Variational Principle ◽  
Critical Points ◽  
Ekeland Variational Principle ◽  
Minimax Theorems

We prove two minimax principles to find almost critical points ofC1functionals restricted to globally definedC1manifolds of codimension1. The proof of the theorems relies on Ekeland variational principle.

scholarly journals SYMPLECTIC CONNECTIONS WITH A PARALLEL RICCI CURVATURE

2003 ◽  
Vol 46 (3) ◽  
pp. 747-766 ◽  
Author(s):  
Charles Boubel
Keyword(s):  
Riemannian Manifold ◽  
Symplectic Manifold ◽  
Ricci Curvature ◽  
Ricci Tensor ◽  
Decomposition Theorem ◽  
Symplectic Connections ◽  
Symplectic Spaces ◽  
Levi Civita Connection ◽  
Parallel Ricci Tensor ◽  
Symplectic Connection

AbstractA symplectic connection on a symplectic manifold, unlike the Levi-Civita connection on a Riemannian manifold, is not unique. However, some spaces admit a canonical connection (symmetric symplectic spaces, Kähler manifolds, etc.); besides, some ‘preferred’ symplectic connections can be defined in some situations. These facts motivate a study of the symplectic connections, inducing a parallel Ricci tensor. This paper gives the possible forms of the Ricci curvature on such manifolds and gives a decomposition theorem (linked with the holonomy decomposition) for them.AMS 2000 Mathematics subject classification: Primary 53B05; 53B30; 53B35; 53C25; 53C55

scholarly journals CHARGE ORBITS OF SYMMETRIC SPECIAL GEOMETRIES AND ATTRACTORS

2006 ◽  
Vol 21 (25) ◽  
pp. 5043-5097 ◽  
Author(s):  
STEFANO BELLUCCI ◽  
SERGIO FERRARA ◽  
MURAT GÜNAYDIN ◽  
ALESSIO MARRANI
Keyword(s):  
Black Hole ◽  
Critical Points ◽  
Symmetric Spaces ◽  
Central Charge ◽  
Scalar Potential ◽  
Scalar Fields ◽  
Extremal Black Hole ◽  
Kahler Geometry ◽  
Complex Scalar ◽  
Black Hole Background

We study the critical points of the black hole scalar potential V BH in N = 2, d = 4 supergravity coupled to nV vector multiplets, in an asymptotically flat extremal black hole background described by a 2(nV+1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold. For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with nonvanishing Bekenstein–Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(nV+1)-dimensional representation RV of the U-duality group. Such orbits are nondegenerate, namely they have nonvanishing quartic invariant (for rank-3 spaces). Other than the ½-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

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