A Splitting Theorem for Alexandrov Spaces
نویسندگان
چکیده
A classical result of Toponogov [12] states that if a complete Riemannian manifold M with nonnegative sectional curvature contains a straight line, thenM is isometric to the metric product of a nonnegatively curved manifold and a line. We then know that the Busemann function associated with the straight line is an affine function, namely, a function which is affine on each unit speed geodesic in the one variable sense. After the theorem, many generalizations were proved. Cheeger-Gromoll’s theorem [2] is the most excellent one among them. An Alexandrov space with curvature bounded below by κ ∈ R is a locally compact, complete and path connected inner metric space on which the triangle comparison theorem holds (see [1]). For simplicity, we denote by curv ≥ κ the lower curvature bound. The direct generalization of the Toponogov theorem for Alexandrov spaces with curv ≥ 0 was proved early in 1967 by A. Milka [8]. We see that this is essentially implied by the rigidity of geodesic triangles and hinges in the Global Comparison Theorem (see Fact 1.0). The author has shown in [7] that if a 2-dimensional Alexandrov space X with curv ≥ −κ2 without boundary admits a nontrivial affine function, then X is isometric to flat R2 or flat S1×R. In the present paper, we extend this to higher dimensional Alexandrov spaces, possibly with nonempty boundary, admitting affine functions with a new notion of differentiability. Innami [6] showed that every complete Riemannian manifold admitting a nontrivial affine function splits isometrically into the metric product of a line and a Riemannian manifold. Affine functions on complete Riemannian manifolds naturally possess the differentiability introduced in this paper. We shall define some notion needed to state our main theorem. Let X be an n-dimensional Alexandrov space with curv ≥ −κ2 and n ≥ 2, κ > 0. We denote by pq a minimal geodesic from p to q and by |p, q| the distance
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تاریخ انتشار 2002