Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold $M$, denoted respectively by $\delta (n_{1},\ldots ,n_{k})$ and $\hat \delta (n_{1},\ldots ,n_{k})$, which trivially satisfy $\delta (n_{1},\ldots ,n_{k})\geq \hat \delta (n_{1},\ldots ,n_{k})$. In this article, we completely determine the Riemannian manifolds satisfying the condition $\delta (n_{1},\ldots ,n_{k})=\hat \delta (n_{1},\ldots ,n_{k})$. By applying the notions of these $\delta$-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.