Generalized Toponogov’s theorem for manifolds with radial curvature bounded below. (English) Zbl 1046.53017

Itokawa, Yoe; Machigashira, Yoshiroh; Shiohama, Katsuhiro

Bland, John (ed.) et al., Explorations in complex and Riemannian geometry. A volume dedicated to Robert E. Greene. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3273-5/pbk). Contemp. Math. 332, 121-130 (2003).

The Toponogov comparison theorem for geodesic triangles plays an important role in the investigation of the curvature and topology of Riemannian manifolds. In the classical version of this theorem the comparison manifold is a simply connected surface of constant curvature. U. Abresch [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 651–670 (1985; Zbl 0595.53043)] and D. Elerath [J. Differ. Geom. 15, 187–216 (1980; Zbl 0526.53043)] have proved versions of the Toponogov theorem in which the comparison manifold is a more general surface. Here the authors prove further results of this type, extending the work of Abresch and Elerath. Topological consequences of the results will appear elsewhere.
For the entire collection see [Zbl 1023.00007].

Reviewer: John Urbas (Canberra)

MSC:

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

Keywords:

Toponogov comparison theorem; radial curvature function

Citations:

Zbl 0595.53043; Zbl 0526.53043