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 n > 3 is diffeomorphic to a Euclidean sphere or a real projective space if the inequality Ric_max(x) < nK_min(x) g is true at each point x ∈ M, where Ric_max(x) is the maximum of the Ricci curvature, and K_min(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.