By Michèle Audin

The cloth and references during this prolonged moment version of "The Topology of Torus activities on Symplectic Manifolds", released as quantity ninety three during this sequence in 1991, were up to date. Symplectic manifolds and torus activities are investigated, with a variety of examples of torus activities, for example on a few moduli areas. even supposing the ebook remains to be established on convexity effects, it comprises even more fabric, particularly plenty of new examples and routines.

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The interested readers may consult [24] for a rather complete list of definitions and references. 4. Appendix: Lie groups, Lie algebras, homogeneous spaces Recall that a group G is said to be a Lie group if this is both a group and a manifold, the two structures being compatible in the sense that the group operations are differentiable mappings. This structure is particularly interesting, because it allows one to use the infinitesimal calculus. The main examples are the ones used in this book, the (compact) torus Tn = Rn;zn, the (complex) torus (C*)n, the linear groups GL(n; R), GL(n; C) and their subgroups, O(n), U(n), and so on.

Calibrated almost complex structures Recall that en is a model for the symplectic vector spaces of dimension 2n. This is not accidental: on any symplectic manifold, there are almost complex structures. 2. CALIBRATED ALMOST COMPLEX STRUCTURES 53 A (linear) complex structure on a real vector space E is an endomorphism J of E such that J2 = -1. Analogously, an almost complex structure on a manifold W is a section J of the bundle End TW such that f; = - IdTxW for all x in W. For instance, if W is a complex manifold (namely a manifold with complex holomorphic change of local coordinates), its tangent space Tx W at any point has a natural structure of complex vector space and the multiplication by i is an almost complex structure.

Notice that it follows from the proposition that there is a whole neighborhood of 1 in the image of the exponential mapping. From this the next proposition is deduced. 5. In a connected Lie group, any element can be written as a product of exponentials. 4. APPENDIX: LIE GROUPS, LIE ALGEBRAS, HOMOGENEOUS SPACES 35 Proof A connected Lie group is path-connected. Let g be any element of the group G and let "( : [0, 1] ---+ G be a path from 1 to g. Let U be an open neighborhood of 1 that is included in the image of the exponential mapping.