On modified Einstein tensors and two smooth invariants of compact manifolds

Let $(M,g)$ be a Riemannian $n$-manifold, we denote by $\Ric$ and $\Scal$ the Ricci and the scalar curvatures of $g$. For scalars $k<n$, the modified Einstein tensors denoted $\Eink$ are defined as $\Eink :=\Scal \, g -k\Ric$. Note that the usual Einstein tensor coincides with the half of $\Eint$ and ${\rm Ein}_0=\Scal.g$. It turns out that all these new modified tensors, for $0<k<n$, are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. In this paper we study the positivity properties of these tensors that generalize the positivity properties of the scalar curvature ($k=0$) and positive Einstein curvature ($k=2$). The positivity of $\Eink$ for some positive $k$ implies the positivity of all ${\rm Ein}_l$ with $0\leq l\leq k$ and so we define a smooth invariant $\cEin(M)$ of $M$ to be the supremum of positive k's that renders $\Eink$ positive. By definition $\cEin(M)\in [0,n]$, it is zero if and only if $M$ has no positive scalar curvature metrics and it is maximal equal to $n$ if $M$ possesses an Einstein metric with positive scalar curvature. In some sense, $\cEin(M)$ measures how far is $M$ to admit an Einstein metric of positive scalar curvature. In this paper we prove that $\cEin(M)\geq 2$ if $M$ admits an effective action by a non abelian connected Lie group or if $M$ is simply connected of positive scalar curvature and dimension $\geq 5$. We prove as well that the invariant $\cEin$ increases after a surgery operation on the manifold $M$ or by assuming that the manifold $M$ has higher connectivity. We prove that the condition $\cEin(M)\leq n-2$ does not imply any restriction on the first fundamental group of $M$. We define and prove similar properties for an analogous invariant namely $\cein(M)$. The paper contains several open questions.