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Topology of complete noncompact manifolds. (English) Zbl 0551.53021

Geometry of geodesics and related topics, Proc. Symp., Tokyo 1982, Adv. Stud. Pure Math. 3, 423-450 (1984).
[For the entire collection see Zbl 0535.00016.]
This article surveys structure and splitting theorems of complete noncompact Riemannian manifolds which rely on the existence of (nontrivial) convex functions in general or on the presence of Busemann functions in particular, mainly in the case of nonnegative sectional curvature. The author aims at avoiding complicated comparison theorems and thus gives more direct and elementary proofs, including those of the soul theorem of Cheeger-Gromoll, the splitting theorem of Toponogov and the gap theorem of Greene-Wu. An essential tool are locally nonconstant convex functions, as commonly developed by R. E. Greene and the author [Invent. Math. 63, 129-157 (1981; Zbl 0468.53033) and Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 357-367 (1981; Zbl 0488.57012)]. At the end, the geometric meaning of total curvature is discussed for surfaces, and similar questions are posed for the higher dimensional case in connection with the reviewer’s inequality of the Cohn-Vossen type [J. Differ. Geom. 10, 167-180 (1975; Zbl 0308.53042)].
For the topics, alluded to above, the article supplements the surveys of J. D. Burago and V. A. Zalgaller [Russ. Math. Surv. 32, 1-57 (1977; Zbl 0397.53031)] and the reviewer [Jahresber. Dtsch. Math.-Ver. 83, 1-31 (1981; Zbl 0469.53034)] by including recent papers which appeared since then.
Reviewer: R.Walter

MSC:

53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry