By Michèle Audin
This is a longer moment variation of "The Topology of Torus activities on Symplectic Manifolds" released during this sequence in 1991. the cloth and references were up to date. Symplectic manifolds and torus activities are investigated, with a number of examples of torus activities, for example on a few moduli areas. even though the publication continues to be founded on convexity theorems, it comprises even more effects, proofs and examples.
Chapter I bargains with Lie crew activities on manifolds. In Chapters II and III, symplectic geometry and Hamiltonian staff activities are brought, particularly torus activities and action-angle variables. The center of the e-book is bankruptcy IV that is dedicated to functions of Morse concept to Hamiltonian staff activities, together with convexity theorems. As a kin of examples of symplectic manifolds, moduli areas of flat connections are mentioned in bankruptcy V. Then, bankruptcy VI facilities at the Duistermaat-Heckman theorem. In bankruptcy VII, a topological development of complicated toric kinds is gifted, and the final bankruptcy illustrates the brought equipment for Hamiltonian circle activities on 4-manifolds.
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Additional info for Torus Actions on Symplectic Manifolds
Let y D y i Ag i be an arbitrary vector in En . 130). Then, Ax D y i Ag i D y which implies that the tensor A is inverse to A 1 . 131) implies the uniqueness of the inverse. Indeed, if A 1 and A 1 are two distinct tensors both inverse to A then there exists at least one vector y 2 En such that A 1 y ¤ A 1 y. 131) into account we immediately come to the contradiction. 130) x D B 1 A 1 y; 8x 2 En : On the basis of transposition and inversion one defines the so-called orthogonal tensors. 73) is orthogonal.
Its elements are second-order tensors that can be treated as vectors in En with all the operations specific for vectors such as summation, multiplication by a scalar or a scalar product (the latter one will be defined for second-order tensors in Sect. 10). 9 Special Operations with Second-Order Tensors 21 for second-order tensors one can additionally define some special operations as for example composition, transposition or inversion. Composition (simple contraction). Let A; B 2 Linn be two second-order tensors.
T/ : dt dt dt 2. t/ : dt dt dt M. 5) 35 36 2 Vector and Tensor Analysis in Euclidean Space 3. t/ W : dt dt dt 4. 8) 5. 9) 6. t/ D : dt du dt dt du dt 7. 12) dt @u dt @v dt where @=@u denotes the partial derivative. 0 @u s The above differentiation rules can be verified with the aid of elementary differential calculus. 9) we proceed as follows. 1. x 1 ; x 2 ; : : : ; x n /. These numbers are called coordinates of the corresponding vectors. i D 1; 2; : : : ; n/. r/ and r D r x 1 ; x 2 ; : : : ; x n are sufficiently differentiable.
Torus Actions on Symplectic Manifolds by Michèle Audin