On modified Einstein tensors and two smooth invariants of compact manifolds

Abstract

Let ( M , g ) (M,g) be a Riemannian n n -manifold, we denote by R i c Ric and S c a l Scal the Ricci and the scalar curvatures of g g . For each real number k > n k>n , the modified Einstein tensors denoted E i n k \mathrm {Ein}_k is defined to be E i n k ≔ S c a l g − k R i c \mathrm {Ein}_k ≔Scal\, g -kRic . Note that the usual Einstein tensor coincides with one half of E i n 2 \mathrm {Ein}_2 and E i n 0 = S c a l . g \mathrm {Ein}_0=Scal.g . It turns out that all these new modified tensors, for 0 > k > n 0>k>n , are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. The positivity of E i n k \mathrm {Ein}_k for some positive k k implies the positivity of all E i n l \mathrm {Ein}_l with 0 ≤ l ≤ k 0\leq l\leq k and so we define a smooth invariant E i n ( M ) \mathbf {Ein}(M) of M M to be the supremum of positive k’s that renders E i n k \mathrm {Ein}_k positive. By definition E i n ( M ) ∈ [ 0 , n ] \mathbf {Ein}(M)\in [0,n] , it is zero if and only if M M has no positive scalar curvature metrics and it is maximal equal to n n if M M possesses an Einstein metric with positive scalar curvature. In some sense, E i n ( M ) \mathbf {Ein}(M) measures how far M M is away from admitting an Einstein metric of positive scalar curvature. In this paper, we prove that E i n ( M ) ≥ 2 \mathbf {Ein}(M)\geq 2 , for any closed simply connected manifold M M of positive scalar curvature and with dimension ≥ 5 \geq 5 . Furthermore, for a compact 2 2 -connected manifold M M with dimension ≥ 6 \geq 6 and of positive scalar curvature, we show that E i n ( M ) ≥ 3 \mathbf {Ein}(M)\geq 3 . We demonstrate as well that the invariant E i n ( M ) \mathbf {Ein} (M) of a manifold M M increases after doing a surgery on M M or by assuming that M M has higher connectivity. We show that the condition E i n ( M ) ≤ n − 2 \mathbf {Ein}(M)\leq n-2 does not imply any restriction on the first fundamental group of M M . We define and prove similar properties for an analogous invariant namely e i n ( M ) \mathbf {ein}(M) . The paper contains several open questions.