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Is the holonomy group of the cotangent bundle, of a compact riemannian manifold, with respect to te standard symplectic structure equal to $SU(n)$, where $n$ is the dimension of the riemannian manifold?


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up vote 4 down vote accepted

No. Of course, we must first assume $\mathcal{M}$ is non-flat to have any chance. Then, while it is true that the Sasaki metric on the tangent bundle $T\mathcal{M}$ along with the canonical symplectic structure form an almost-Hermitian structure, its torsion need not vanish. Even when it does, the holonomy group may not equal $SU_n$; the tangent bundles of complex projective spaces are hyper-Kaehler.

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