2026/07/20 by Jian Ge
Mathematics · #math.DG #math.GT
Let V be an n-dimensional Euclidean vector space, ,where n≥ 4, and ℓ = \lfloor(n)/(2)\rfloor. We prove the sharp pointwise estimate q2(E) ≥ -(2(ℓ -1))/(3ℓ) Scal(E) IdΛ2V^* for every algebraic curvature tensor E on V with nonnegative sectional curvature. Applying this estimate to the decomposition Rmg=KminI+E, we obtain the vanishing of H2(M; ℝ) under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields b2(M)=0 in odd dimensions and b2(M)≤ 1 in even dimensions. At even-dimensional endpoint, b2(M)>0 forces (M, g) to be isometric, up to scaling, to \mathbbCPℓ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion Kmin ≥ (n(n-1))/(n2-n+12)S0 \Longrightarrow PIC2. The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.