Transformation techniques for partial differential equations on projectively flat manifolds (Q1895227)

The present paper is devoted to the development and extension of transformation techniques between Codazzi tensors and Codazzi operators introduced by \textit{F. Dillen}, \textit{K. Nomizu} and \textit{L. Vrancken} [Monatsh. Math. 109, No. 3, 221-235 (1990; Zbl 0712.53008)]. These techniques are useful in the study of the relation between conformal and projective structures on the one hand and certain second-order PDE's on the other. Let \(M\) be diffeomorphic to \(S^n(1)\) and \(\nabla\) be an affine connection without torsion and with symmetric Ricci tensor. For \(f\in C^\infty(M)\) define the tensor field \[ H(f):= \text{Hess } f+ {1\over n- 1} f\cdot\text{Ricc}, \] where \(\text{Hess } f\) denotes the covariant Hessian of \(f\) relative to \(\nabla\). For a given Riemannian metric \(h\) on \(M\) and a given torsion free, Ricci- symmetric, projectively flat connection \(\nabla^*\) define the linear second-order operator \(D(h, \nabla^*)(f):= \text{trace}_h H(f)\). The author studies the solvability of the equations \[ \begin{aligned} D(h, \nabla^*)(f) & = 0\tag{1}\\ \text{and} D(h, \nabla^*)(f) & = \varphi,\quad \varphi\in C^\infty(M).\tag{2}\end{aligned} \] Assuming that \(\nabla^*\) satisfies the Codazzi equation \((\nabla^*_u h)(v, w)= (\nabla^*_v h)(u, w)\) he establishes some necessary and sufficient conditions for the existence of solutions of the equations (1) and (2). Moreover, if \(f\), \(\widehat f\) are two solutions of (2), then \(f-\widehat f\) is a solution of (1). The rest of the paper contains some consequences and applications of these results: 1) a global representation of Codazzi tensors on projectively flat spaces; 2) an answer to the problem which data computed in terms of the projectively flat connection determine the connection uniquely; 3) some existence and uniqueness theorems for hyperovaloids in terms of relative differential geometry.