Abstract
Let mathcal {W}^{n} be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let Win mathcal {W}^{n} and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in Win mathcal {W} are considered.
Highlights
Let Wn be the set of smooth complete connected n-dimensional manifolds without conjugate points
The Busemann function bω : W ∈ Wn → R is defined by bω (x)
The right hand side is well-defined and the Busemann function bω is smooth in a complete connected manifold without conjugate point W ∈ Wn whereas bω is at least C2 given that W has no focal points(see [1, Theorem 2])
Summary
Let Wn be the set of smooth complete connected n-dimensional manifolds without conjugate points. The right hand side is well-defined and the Busemann function bω is smooth in a complete connected manifold without conjugate point W ∈ Wn whereas bω is at least C2 given that W has no focal points(see [1, Theorem 2]).
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