On modified Einstein tensors and two smooth invariants of compact manifolds

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