Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature

In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangentia