Notes on 4-dimensional hyper-para-Kähler manifolds (Q1763116)

Let \((M, I, J, K, g)\) be a \(4\)-dimensional almost hyper-Hermitian manifold and \(R\) its Riemannian curvature. \(M\) is called hyper-para-Kähler if \(M\) satisfies the conditions \[ R(X, Y) \cdot I = R(X, Y) \cdot J = R(X, Y) \cdot K = 0, \] for any vector fields \(X, Y\) on \(M\). In the present paper the author proves that a \(4\)-dimensional hyper-para-Kähler manifold \((M, I, J, K, g)\) is locally deformable to a hyper-Kähler manifold, under a local smooth SO(3)-valued function.