every symplectic manifold has even dimension

All we need to prove is that every finite dimensional vector spaceMathworldPlanetmath V with an anti-symmetric non-degenerate linear formPlanetmathPlanetmath ω has an even dimensionPlanetmathPlanetmath 2k. This is only a linear algebraMathworldPlanetmath result. In the case of a symplectic manifold V is just the tangent space at a point, and thus its dimension equals the manifold’s dimension.

Pick any not null vector v0V. Since ω is non-degenerate ω(v0,) is a not null linear form. Therefore there exists a not null vector u0 such that ω(v0,u0)=1

Now v0 and u0 are linearly independentMathworldPlanetmath because if v0=λu0 then ω(v0,u0)=ω(λu0,u0)=λω(u0,u0)=0 (by anti-symmetry).

Let V0=span{v0,u0}. Consider a space V1 of ”orthogonalMathworldPlanetmath” elements to V0 under ω. That is:

V1={v1V: for all vV0,ω(v,v1)=0}

We now prove V=V0V1:

  • V0V1={0}

    Suppose wV0V1 is not null, then it can be written w=αv0+βu0 because it belongs to V0. Since it also belongs to V1 is is ”orthogonal” to both v0 and u0. That is:

    ω(v0,w)=0βω(v0,u0)=0β=0 similarly


    So w must be null.

  • V=V0V1

    Suppose wV. Let α=ω(v0,w), β=ω(u0,w), w0=αu0-βv0.

    Then ω(v0,w0)=α=ω(v0,w) and ω(u0,w0)=β=ω(u0,w).

    Considering w1=w-w0 we have w=w0+w1 (by construction) and ω(v0,w1)=ω(v0,w-w0)=ω(v0,w)-ω(v0,w0)=ω(v0,w)-ω(v0,w)=0 and similarly for ω(u0,w1)

    So w1V1, w0V0 and w=w0+w1 and thus V=V0V1

So the matrix representationPlanetmathPlanetmath of ω is block-diagonal in V0V1 and a restriction anti-symmetric bilinearPlanetmathPlanetmath for of ω to V1 exists.

If V1 is not null we can repeat the procedure with the restriction. Since dim(V)=dim(V0)+dim(V1) and V is finite dimensional the procedure must stop at a finite step.

At the end we get a decomposition V=i=0k-1Vi, where dim(Vi)=2 and dim(V)=2k is even.

Title every symplectic manifold has even dimension
Canonical name EverySymplecticManifoldHasEvenDimension
Date of creation 2013-03-22 15:44:05
Last modified on 2013-03-22 15:44:05
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 18
Author cvalente (11260)
Entry type Theorem
Classification msc 53D05
Related topic AlternatingForm