Showing papers on "Symplectic vector space published in 1976"

Some simple examples of symplectic manifolds

Abstract: This is a construction of closed symplectic manifolds with no Kaehler structure. A symplectic manifold is a manifold of dimension 2k with a closed 2-form a such that ak is nonsingular. If M2k is a closed symplectic manifold, then the cohomology class of a is nontrivial, and all its powers through k are nontrivial. M also has an almost complex structure associated with a, up-to homotopy. It has been asked whether every closed symplectic manifold has also a Kaehler structure (the converse is immediate). A Kaehler manifold has the property that its odd dimensional Betti numbers are even. H. Guggenheimer claimed [1], [2] that a symplectic manifold also has even odd Betti numbers. In the review [3] of [1], Liberman noted that the proof was incomplete. We produce elementary examples of symplectic manifolds which are not Kaehler by constructing counterexamples to Guggenheimer's assertion. There is a representation p of Z E Z in the group of diffeomorphisms of T2 defined by (1, 0) -P4 id, (0,1 I 0o 1l where [81 ]" denotes the transformation of T2 covered by the linear transformation of R2. This representation determines a bundle M4 over T with fiber T2: M4 = T2 XZ9Z T2, where Z E Z acts on T2 by covering transformations, and on T2 by p (M4 can also be seen as R4 modulo a group of affine transformations). Let Q1 be the standard volume form for T2. Since p preserves 21, this defines a closed 2-form i2 on M4 which is nonsingular on each fiber. Let p be projection to the base: then it can be checked that S21 + P*' 1 is a symplectic form. (It is, in general, true that "'j + Kp* 21 is a symplectic form, for any closed Q'1 which is a volume form for each fiber, and K sufficiently large.) But H1 (M4) = Z @ Z @ Z, so M4 is not a Kaehler manifold. Many more examples can be constructed. In the same vein, if M2k is a closed symplectic manifold, and if N2k+2 fibers over M2k with the fundamental class of the fiber not homologous to zero in N, then N is also a symplectic manifold. If, for instance, the Euler characteristic of the fiber is not zero, this Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 57D15, 58H05. C American Mathematical Society 1976

Shorter Notes: Some Simple Examples of Symplectic Manifolds

Abstract: This is a construction of closed symplectic manifolds with no Kaehler structure. A symplectic manifold is a manifold of dimension 2k with a closed 2-form a such that ak is nonsingular. If M2k is a closed symplectic manifold, then the cohomology class of a is nontrivial, and all its powers through k are nontrivial. M also has an almost complex structure associated with a, up-to homotopy. It has been asked whether every closed symplectic manifold has also a Kaehler structure (the converse is immediate). A Kaehler manifold has the property that its odd dimensional Betti numbers are even. H. Guggenheimer claimed [1], [2] that a symplectic manifold also has even odd Betti numbers. In the review [3] of [1], Liberman noted that the proof was incomplete. We produce elementary examples of symplectic manifolds which are not Kaehler by constructing counterexamples to Guggenheimer's assertion. There is a representation p of Z E Z in the group of diffeomorphisms of T2 defined by (1, 0) -P4 id, (0,1 I 0o 1l where [81 ]" denotes the transformation of T2 covered by the linear transformation of R2. This representation determines a bundle M4 over T with fiber T2: M4 = T2 XZ9Z T2, where Z E Z acts on T2 by covering transformations, and on T2 by p (M4 can also be seen as R4 modulo a group of affine transformations). Let Q1 be the standard volume form for T2. Since p preserves 21, this defines a closed 2-form i2 on M4 which is nonsingular on each fiber. Let p be projection to the base: then it can be checked that S21 + P*' 1 is a symplectic form. (It is, in general, true that "'j + Kp* 21 is a symplectic form, for any closed Q'1 which is a volume form for each fiber, and K sufficiently large.) But H1 (M4) = Z @ Z @ Z, so M4 is not a Kaehler manifold. Many more examples can be constructed. In the same vein, if M2k is a closed symplectic manifold, and if N2k+2 fibers over M2k with the fundamental class of the fiber not homologous to zero in N, then N is also a symplectic manifold. If, for instance, the Euler characteristic of the fiber is not zero, this Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 57D15, 58H05. C American Mathematical Society 1976

Symplectic Manifolds, Dynamical Groups, and Hamiltonian Mechanics

Abstract: Let (M, ω) be a symplectic manifold and G a Lie group acting on M by symplectic diffeomorphisms (i.e. g*ω = ω for all g ∈ G). Souriau has defined the moment of this group action as a suitable map J : M → G* (dual of the Lie algebra G of G). Some well-known results are first briefly outlined in Part I: there exists an action of G on G* for which J is equivariant, whose orbits ϑξ are symplectic manifolds (Kirillov-Souriau-Kostant’s theorem); if ξ is a regular value of J, J −1(ξ) quotiented by the action of the isotropy subgroup of ξ is, under suitable assumptions, a symplectic manifold (Meyer’s theorem).

A symplectic structure on the set of Einstein metrics. A canonical formalism for general relativity

Abstract: A symplectic structure i.e. a symplectic form Γ on the set of all solutions of the Einstein equations on a given 4-dimensional manifold is defined. A degeneracy distribution of Γ is investigated and its connection with an action of the diffeomorphism group is established. A multiphase formulation of General Relativity is presented. A superphase space for General Relativity is proposed.

Eigenvalues of the Casimir operators of the orthogonal and symplectic groups

Abstract: Eigenvalues of the Casimir operators of the orthogonal and the symplectic groups are obtained in closed and simple form by diagonalizing directly the matrices introduced by Perelomov and Popov This method unifies the treatment of the problem for the semisimple Lie groups

An example of moduli for singular symplectic forms

Abstract: In [3] Mar t ine t shows that there are four generic types of s ingulari t ies for germs of closed C ~ 2-forms on 4-manifolds and then defines a no t ion of s tabil i ty for these germs. The stabi l i ty of the first s ingular i ty type is just the classical D a r b o u x theorem for symplect ic forms. Mar t ine t proved the s tabi l i ty of the second type; while, more recently, Roussar ie [6] has shown the s tabi l i ty of the third. In this paper we shall show that forms exhibi t ing this last type of s ingular i ty are unfor tunate ly not stable. In fact, we show that near any generic X2.2.1 singular i ty there is, at least, a one pa rame te r family of moduli . In w we briefly descr ibe the var ious singularit ies. In w we will show how to reduce the p rob lem of s tabi l i ty to one involving a contact s t ructure on IR 3 at 0. Section 3 conta ins the p r o o f of instabi l i ty . Note : we assume that all functions, forms, vector fields, etc. are C ~.

Canonical realizations of the lie algebra sp(2 n, R)

Abstract: The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp i andq i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.

Some applications of Landweber-Novikov operations

Abstract: Previous results on the characteristic numbers of Sp-manifolds are extended in three different ways. I. It is shown that the primitive symplectic Pontrjagin class evaluated on a 4(2J 1) dimensional Spmanifold always gives a number divisible by 8. This forms an analogue to a well-known result of Milnor concerning U-manifolds. II. It is shown that some of the results of Floyd as well as an analogue of the previous result can be obtained for 'pseudo-symplectic' manifolds. III. Results are generalised to (Sp,fr) manifolds. 1. 4(2i 1) dimensional Sp-manifolds. Let s,,(p)[M], J a partition of n = n(n), M a 4n(7) dimensional stably symplectic manifold, denote the normal symplectic Pontrjagin number of M corresponding to the 7Tsymmetrised polynomial in a system of indeterminates for which the symplectic Pontrjagin classes are the elementary symmetric polynomials. Throughout this section we will set k = 1 and M will denote a 4k dimensional stably symplectic manifold. THEOREM 1.1. 8 | S(k)(p)[M]. REMARKS. 1. The unitary analogue, 2 s(k)(c)[N], N stably unitary is well known; it could be proven by the techniques used below. 2. The techniques of [3] are not adequate by themselves to prove Theorem 1.1. PROOF. Actually we will prove slightly more: Let 7T be any partition of k all of whose parts are themselves integers of the form 25 1. Then 8 I s,(p)[M]. If r =(a,, ... , ar), let D(X) = 11[(2ai + 2)!/2]. Well known fact. 2 1 2fn(,l)=k(s,(p)[M]/D(7)). This is the 'Todd genus' relation of Stong [4] who put things in an 'abnormal' form; using normal rather than tangential numbers makes computation manageable. In particular, we can see that for a fixed k the denominators D (7) with maximal number of factors of 2 will be just those for which all parts of 7 are of the form 25 1. By Proposition 4 of [3] it is automatic that 4 s,,(p)[M] for all ', n(r) = k. If we can show that 8 I s,(p)[M] whenever , = (a,, . .. , ar), n(rr) = k, r > 1 Presented to the Society, Jaruary 26, 1975; received by the editors November 1, 1974 and, in revised form, January 24, 1975. AMS (MOS) subject classifications (1970). Primary 57A70; Secondary 55B20.

_{}actions on symplectic manifolds

Abstract: A bordism classification is studied for periodic maps of prime period p preserving a symplectic structure on a smooth manifold. In sharp contrast to the corresponding oriented bordism, this theory contains nontrivial ptorsion even when p is odd. Calculation gives an upper limit on the size of this p-torsion.