A New Differentiable Sphere Theorem and Its Applications

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold (M, g) of even dimension n3 is diffeomorphic to a Euclidean sphere or a real projective space if the inequality Ric ₌₀ₗ (x) n K ₌₈₍ (x) g is true at each point x M, where Ric ₌₀ₗ (x) is the maximum of the Ricci curvature, and K ₌₈₍ (x) is the minimum of the sectional curvature of (M, g) at x. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.