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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On modified Einstein tensors and two smooth invariants of compact manifolds
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by Mohammed Larbi Labbi HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 1717-1738 Request permission


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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Additional 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:
  • 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:
  • MathSciNet review: 4549690