## radially half-complete projective manifold

S.Choi in his article " Geometric structures on low-dimensional manifolds " introduced some new concepts.one of them is " radially half-complete projective manifold ". In section 2 , paragraph 1 in the above article , he said : " a projective manifold is said to be radially half complete if each point of the universal cover has a neighborhood which mapped homeomorphically to an open cone with a common vertex by a developing map where the vertex is independent of the neighborhoods but may be dependent on developing maps.obviously a radially half-complete projective manifold is diffeomeorphic to a surface times an open interval."

what does " independent of neighborhoods but may dependent on developing maps " mean here? why a radially half-complete projective manifold is diffeomorphic to a surface times an open interval?

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