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Let M be a compact simply connected riemannian manifold of even dimension d . It is well known that if the sectional curvature of M lies in the range (0, A], then M has volume greater than or equal to that of the d-dimensional euclidean sphere Sd of constant curvature A. We prove that d if the volume of M is no greater than 3/2 times that of Sd, then M is homeomorphic with the sphere.

A sphere theorem for reverse volume pinching on even-dimensional manifolds