Existence of metrics with prescribed Ricci curvature: Local theory

The Cauchy-Kowalevski theorem applied for counting connections with a prescribed Ricci tensor

“…The question of existence of connections with a prescribed Ricci tensor was studied, for instance, in [1,2,4,5]. In particular, it was proved in [5] that if n ≥ 2 then any analytic symmetric tensor of type (0, 2) on an n-manifold can be locally realized as the symmetric part of the Ricci tensor of some torsion-free connection.…”

A note on Ricci signatures

“…For the sake of simplicity, we assume that d = 1. Then we can express e 4 as a linear combination of {e 1 With these brackets in hand, it is tedious but straightforward to compute the Ricci curvature R ij . The results are as follows:…”

“…However, there seems to be little knowledge about Ricci signatures with mixed signs. Here, we just want to mention D. M. DeTurck's paper [1], which implies the local solvability of the prescribed Ricci signature problem in the absence of zeroes.…”

Einstein manifolds with convex boundaries

“…The "classical" solution (see [DeT81,GL91,Bes87,Biq00]) is to introduce another, elliptic non-degenerate problem, and to show that its solutions are actually solutions of the original problem. This should be done here with some care regarding the boundary conditions.…”