4 PAUL E. EHRLICH AND KEVIN L. EASLEY upper semicontinuous in a topology of uniform convergence. As part of this work, Busemann studied finite compactness and metric completeness in the spirit of (1) and ( 4) of the Hopf-Rinow Theorem, as recalled in Section 1. In Beem's first co-authored work, Busemann and Beem [40], undertaken before Beem started work on his thesis, aspects of generalized indefinite metric spaces were studied. In particular, the authors offered two basic examples of such spaces- Minkowski space on the first page of their paper, and what has become known as the Lorentz-Poincare model, discussed in the last two pages of their paper. Recall that in basic complex variables or differential geometry, if (2.1) U = {(x,y) E 1R2 : y 0} = lR X (O,+oo) then the Riemannian metric (2.2) furnishes U with a geodesically complete, constant curvature metric with K = -1. If desired, by a change of variables this could be recast as a warped product metric on JR2 . In [40], the set U of (2.1) was retained, but for this second class of examples, the Lorentzian metric (2.3) was considered. It was noted that this Lorentz analogue to the Poincare upper half-plane, while of constant curvature, fails to be geodesically complete. Later, in Beem [14], a more detailed study would be made of this type of Lorentz space. In O'Neill's text (Example 41, p. 209 of [78]), this model is given as a fun- damental example, but in a form which does not on the face of it resemble (2.1) and (2.3). O'Neill has been discussing the geodesic equations for a general semi- Riemannian warped product metric and obtains the helpful result that for any pair of complete Riemannian factors and any warping function, a Riemannian warped product is automatically geodesically complete. To fit the Busemann-Beem exam- ple into this framework, first make a change of variables t = ln y : (0, +oo) ---- JR, so that Now make a change of variables t ~---+ -t, and one has the Lorentzian metric on the manifold M = JRx 1R In Example 41, O'Neill observes that this is an impor- tant metric because while the factors (JR, -dt2 ) and (JR, dx2 ) have complete definite
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