On modified Einstein tensors and two smooth invariants of compact manifolds
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- by Mohammed Larbi Labbi;
- Trans. Amer. Math. Soc. 376 (2023), 1717-1738
- DOI: https://doi.org/10.1090/tran/8791
- Published electronically: October 28, 2022
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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.
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Bibliographic Information
- Mohammed Larbi Labbi
- Affiliation: Department of Mathematics, College of Science, University of Bahrain, 32038, Bahrain
- MR Author ID: 352286
- ORCID: 0000-0003-2960-5372
- Email: mlabbi@uob.edu.bh
- Received by editor(s): February 13, 2022
- Received by editor(s) in revised form: July 25, 2022, and July 27, 2022
- Published electronically: October 28, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 1717-1738
- MSC (2020): Primary 53C20, 53C21
- DOI: https://doi.org/10.1090/tran/8791
- MathSciNet review: 4549690