Linear Complex Structure - Compatibility With Other Structures

Compatibility With Other Structures

If B is a bilinear form on V then we say that J preserves B if

B(Ju, Jv) = B(u, v)

for all u,v in V. An equivalent characterization is that J is skew-adjoint with respect to B:

B(Ju, v) = −B(u, Jv)

If g is an inner product on V then J preserves g if and only if J is an orthogonal transformation. Likewise, J preserves a nondegenerate, skew-symmetric form ω if and only if J is a symplectic transformation (that is, if ω(Ju,Jv) = ω(u,v)). For symplectic forms ω there is usually an added restriction for compatibility between J and ω, namely

ω(u, Ju) > 0

for all u in V. If this condition is satisfied then J is said to tame ω.

Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form g_J on V_J

g_J(u,v) = ω(u,Jv).

Because a symplectic form is nondegenerate, so is the associated bilinear form. Moreover, the associated form is preserved by J if and only if the symplectic form and if ω is tamed by J then the associated form is positive definite. Thus in this case the associated form is a Hermitian form and V_J is an inner product space.