Contemporary Mathematics Volume 359, 2004 A Beemian Sampler: 1966-2002 Paul E. Ehrlich and Kevin L. Easley ABSTRACT. A survey is given of aspects of the development of global space-time differ- ential geometry since the 1970s, focusing on some major areas of investigation of Professor John Beem, various co-authors, and Ph.D. students. 1. Introduction In accordance with the overall theme of the Beemfest, Advances in Differ- ential Geometry and General Relativity, a selective summary is presented of the research of Professor John Beem and various co-authors and students, beginning with Beem's first joint publication in 1966. Since a time period of nearly four decades is covered, results discussed are placed in historical context. As an aspect of this viewpoint, the language employed in the original papers is often used. Any student of Riemannian geometry is exposed to a wonderful global result and basic working tool, which goes back to Hopf and Rinow [65]. If (N,g0 ) is a Riemannian manifold, then an associated Riemannian distance function do : N x N ---+ JR. is given by do(p, q) = inf{L(c) I c: [0, 1] ---+ N is a piecewise smooth curve with c(O) = p, c(1) = q}. Then the promised result guarantees the equivalence of the following conditions: HoPF-RINOW THEOREM. For any Riemannian manifold (N,go), the following are equivalent: 1. metric completeness: (N, d0 ) is a complete metric space 2. geodesic completeness: for any v E TN, the geodesic cv(t) in N with initial condition cv'(O) = v is defined for all values of an affine parameter t 3. for some point p E N, the exponential map expp is defined on all of TpN 4. finite compactness: every subset K of N that is do-bounded has compact closure. 1991 Mathematics Subject Classification. Primary 53C50, 01A65. © 2004 American Mathematical Society http://dx.doi.org/10.1090/conm/359/06552
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