Journal of the Korean Mathematical Society (대한수학회지)
- Volume 32 Issue 2
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- Pages.351-361
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- 1995
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- 0304-9914(pISSN)
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- 2234-3008(eISSN)
Complete open manifolds and horofunctions
- Yim, Jin-Whan (Department of Mathematics Korea Advanced Institute of Science and Technology)
- Published : 1995.05.01
Abstract
Let M be a complete open Riemannian manifold. When the sectional curvature $K_M$ of M is nonpositive, Gromov has defined, in his lectures [3], the ideal boundary of M, and used it to study the geometric structure of M. In a Hadamard manifold, a simply connected manifold with nonpositive sectional curvature, a point at infinity can be defined as an equivalence class of rays. He proved many interesting theorems using this definition of ideal boundary and the so-called Tit's metric on it. He also suggested a counterpart to this for nonnegative curvature case. This idea has been taken up by Kasue to study the structure of complete open manifolds with asympttically nonnegative curvature [14]. Motivated by these works, we will define an idela boundary of a general noncompact manifold M, and study its structure.