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Generalized space forms. (English) Zbl 0990.53032

Summary: Spaces with radially symmetric curvature at base point \(p\) are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to \({\mathbb R^n}\) or \(S^n\) under a radially symmetric metric, to \({\mathbb R}\text{P}^n\) with Riemannian universal covering of \(S^n\) equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at \(p\).

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
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