On Ricci-Quadratic Sprays (or Finsler) Spaces

In this paper, we study Ricci-quadratic sprays (or Finsler) spaces which are non-trivial in the sense that these sprays (or Finsler) spaces are not strongly Ricci-quadratic. First, we find infinitely many such sprays on an open domain in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ which are not induced by (not necessary positive definite) Finsler metrics. Then, we explicitly construct a lot of non-trivial Ricci-quadratic Finsler metrics by finding the PDE characterization for the spherically symmetric Finsler metrics to be Ricci-quadratic and strongly Ricci-quadratic.