Transcription of MANIFOLDS OF NEGATIVE CURVATURE - American …
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MANIFOLDS OF NEGATIVE CURVATURE - American …
Transactions of the American mathematical society Volume 145, November 1969. MANIFOLDS OF NEGATIVE CURVATURE . BY. R. L. BISHOP AND B. O'NEILLO. 1. Introduction ACm function/on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (/o <t)"5:0 for every geodesic a in M. We shall apply this notion in a variety of ways to the study of MANIFOLDS of NEGATIVE or nonpositive CURVATURE . Convexity has, of course, long been associated with NEGATIVE CURVATURE , but convex functions seem to have been used only locally or along curves. In the first part of this paper we give an abstract global treatment. Nonconstant convex functions exist only on MANIFOLDS of infinite volume ( ); the first question about such a function on M (complete, A' 0) is whether it has a critical point necessarily an absolute minimum.
which uses convex functions to construct a wide variety of manifolds of negative curvature (7.5). For example, negative space forms can easily be constructed in this way from flat space forms (7.10). The study of Killing fields on warped products (8.2) shows in particular that the dimensions and length of mobility sequences are
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