Showing papers on "Symplectic vector space published in 1969"
TL;DR: In this paper, a local version of the Darboux theorem was adapted to prove a similar result for Banach manifolds, where the problem is a local one and the corresponding manifold is a 2-form manifold.
Abstract: 1. Normal form. Let M be a Banach manifold. A symplectic structure on M is a closed 2-form Q such that the associated mapping S: T(M)->T*(M) defined by Q(X) = X _ ] 0 is a bundle isomorphism. If M is finite dimensional, Darboux's theorem states that every point in M has a coordinate neighborhood N with coordinate functions (xi, • • • , xn, yi, • • • , yn) such that Q= ]C?=i dxiAdy% on iV. Standard proofs of this theorem (e.g. [4]) use induction on w, so they do not apply to the infinite-dimensional case. It happens, however, that an idea of J. Moser [3] may be adapted to prove a similar result for Banach manifolds. Since the problem is a local one, it suffices to consider a symplectic structure 0 on a neighborhood of 0 in a Banach space B.
30 citations
01 Jan 1969
TL;DR: In this paper, the additive orders of some important elements in Qs*(Z,) (p prime), the module of principal Zp-manifolds with compatible weak symplectic structure are calculated.
Abstract: The lacunae in our knowledge of the coefficients have made direct computations in the homology theory is?'(X, A) almost impossible This note sidesteps the difficulty by exploiting the relationship between the theories KO* and Qsp, which corresponds to that between KU* and Qu, and which is fully discussed in Chapter II of [1] In this way we are able to calculate the additive orders of some important elements in Qs*(Z,) (p prime), the module of principal Zp-manifolds with compatible weak symplectic structure Perhaps the most interesting aspect of the computation is the difference between the cases p odd and p equals 2, reflecting the presence of 2-torsion in Qsp I am grateful to Elmer Rees for discussing this problem with me and for showing me his version of the computations in [2] for real projective space
1 citations
TL;DR: In this paper, it was shown that if M is a free K [G]-module, then there exists in M a normal basis with a canonical Gram matrix, and that M is also a non-egenerate finite-dimensional symplectic space over K with the matching structure of a G module.
Abstract: Let K be a field of nonzero characteristic p≠2, let G be a finite p-group, and let M be a nondegenerate finite-dimensional symplectic space over K with the matching structure of a Gmodule. It is proven that if M is a free K [G]-module then there exists in M a normal basis with a canonical Gram matrix.