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Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the natural inner product: i.e., ${\mathcal J}: \mathbf{TM}\oplus\mathbf{TM}^*\rightarrow \mathbf{TM}\oplus\mathbf{TM}^*$ such that

${\mathcal J}^2=-{\rm Id},\ \ \mbox{ and }\ \ \langle {\mathcal J}(X+\xi),{\mathcal J}(Y+\eta)\rangle=\langle X+\xi,Y+\eta\rangle$.

Also this inner product can be defined as follows:

If $X$ and $Y$ are vector fields and $ξ$ and $η$ are one-forms then the inner product of $X+ξ$ and $Y+η$ is defined as $\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))$.Note that ${\mathcal J}$ is also symplectic i.e., ${\mathcal J}^*=-{\mathcal J}$.

We know, If $M$ is a manifold equipped with a symplectic structure $\omega$ and an almost complex structure $J$, then $\omega$ and $J$ are said to be compatible if the expression $g(X; Y ) = \omega(X; JY )$ ($\star$), defines a Riemannian metric on $M$. In other words, the bilinear form $g$ must be symmetric and positive definite.

The expression $(\star)$ defines an injective map from the space of compatible almost complex structures to the space of Riemannian metrics on $M$. In fact, one can define a retraction of this map from the space of Riemannian metrics to the space of compatible almost complex structures. Since the space of metrics is convex, the space of compatible almost complex structures is contractible which was proved a priori by M.Gromov . So my question is:

*Question:*The space of generalized complex structures in sense of N.Hitchin is contractible?

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up vote 6 down vote accepted

The space of generalized complex structures at a point of $M$ of dimension $2n$ is diffeomorphic to $O(2n,2n)/U(n,n)$ (see e.g. arxiv:0703298), hence is not even connected.

It has $4$ components, each homotopy equivalent to $SO(2n)/U(n)\times SO(2n)/U(n)$, hence non-contractible as soon as $n>1$.

EDIT: according to Hitchin, the space is rather $SO(2n,2n)/U(n,n)$, which still has two connected components.

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