On modified Einstein tensors and two smooth invariants of compact manifolds

Abstract:

Let $(M,g)$ be a Riemannian $n$-manifold, we denote by $Ric$ and $Scal$ the Ricci and the scalar curvatures of $g$. For each real number $k<n$, the modified Einstein tensors denoted $\mathrm {Ein}_k$ is defined to be $\mathrm {Ein}_k ≔Scal\, g -kRic$. Note that the usual Einstein tensor coincides with one half of $\mathrm {Ein}_2$ and $\mathrm {Ein}_0=Scal.g$. 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. The positivity of $\mathrm {Ein}_k$ for some positive $k$ implies the positivity of all $\mathrm {Ein}_l$ with $0\leq l\leq k$ and so we define a smooth invariant $\mathbf {Ein}(M)$ of $M$ to be the supremum of positive k’s that renders $\mathrm {Ein}_k$ positive. By definition $\mathbf {Ein}(M)\in [0,n]$, 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, $\mathbf {Ein}(M)$ measures how far $M$ is away from admitting an Einstein metric of positive scalar curvature.

In this paper, we prove that $\mathbf {Ein}(M)\geq 2$, for any closed simply connected manifold $M$ of positive scalar curvature and with dimension $\geq 5$. Furthermore, for a compact $2$-connected manifold $M$ with dimension $\geq 6$ and of positive scalar curvature, we show that $\mathbf {Ein}(M)\geq 3$. We demonstrate as well that the invariant $\mathbf {Ein} (M)$ of a manifold $M$ increases after doing a surgery on $M$ or by assuming that $M$ has higher connectivity. We show that the condition $\mathbf {Ein}(M)\leq n-2$ does not imply any restriction on the first fundamental group of $M$. We define and prove similar properties for an analogous invariant namely $\mathbf {ein}(M)$. The paper contains several open questions.