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 $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ 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.