In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. This warped product structure will be used to study warped product Einstein structures in our paper "The space of virtual solutions to the warped product Einstein equation".
He, Chenxu; Petersen, Peter; and Wylie, William, "Warped Product Rigidity" (2011). Mathematics Faculty Scholarship. Paper 132.
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