Holonomy group of cotangent bundle

Question

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?

lorenz

Answer 1

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.