22 PAUL E. EHRLICH AND KEVIN L. EASLEY (F, gp) are complete. One remarkable feature of this result is that completeness holds for any warping function f on B. The Busemann-Beem example described in detail in Section 2 shows that there is no exact Lorentzian analogue of this result, even when both factors are definite, once again illustrating the striking differences between Lorentzian and Riemannian geometry. It is natural to inquire what can be salvaged in the Lorentzian case. The decomposition of the warped product metric g = g EB fh implies that the basic projection maps 1r : M x f H ---+ M and a : M x f H ---+ H have the properties that 1r!Mxb ---+ M is an isometry for each b E H, and aimxH ---+ H is a homothety with factor 1/ f(m) for each mE M. Consequently, given any tangent vector v E T(M x H), one then has the useful inequality g(1r*V,1r*v) :: g(v,v), yielding two immediate consequences: (i) the map 1r* : Tp(M x H) ---+ T1r(p)M maps nonspacelike vectors to nonspacelike vectors, and (ii) the map 1r is length nondecreasing on nonspacelike curves. Within the context of the Robertson-Walker cosmological models, these fun- damental relations were used to obtain the following satisfying Lorentzian analogue for the Riemannian completeness theorem (cf. Beem and Ehrlich [21], p. 65 Beem, Ehrlich, and Easley [24], p. 103) in the simpler product case with base manifold (JR, -dt2 ). THEOREM 7.2. Suppose that (H, h) is a Riemannian manifold and that lR x H is given the product Lorentzian metric -dt2 EB h. Then the following are equivalent: 1. ( H, h) is geodesically complete. 2. (JR x H, -dt2 EB h) is geodesically complete. 3. (JR x H, -dt2 EB h) is globally hyperbolic. If the dimension of the base manifold ( M, g) is two or larger, then the situation naturally becomes more difficult. The following result ( cf. [21], p. 66 [24], p. 104) shows what can be recovered with an additional assumption on the base manifold (M,g). THEOREM 7.3. Let (M, g) be a space-time, and let (H, h) be a Riemannian manifold. Then the Lorentzian warped product (Mx fH, gEBfh) is globally hyperbolic iff both of the following conditions are satisfied: 1. ( M, g) is globally hyperbolic. 2. (H, h) is a complete Riemannian manifold. Along the way to establishing Theorems 7.2 and 7.3, Beem and Ehrlich ([21], p. 61 [24], p. 100) proved the following preliminary result, which is also independent of the warping function. PROPOSITION 7.4. Let (M, g) be a space-time, and let (H, h) be a Riemann- ian manifold. Then the Lorentzian warped product ( M x f H, g EB f h) is strongly causal (respectively, chronological, causal) iff (M, g) is strongly causal (respectively, chronological, causal). Considered together, the above remarks produce a theme of sharp contrasts. In the case of Riemannian warped products with complete base and fiber, geodesic completeness is independent of the choice of warping function f in the space-time case, by contrast, only certain causal aspects of the warped product are independent of the choice of warping function. There is also the additional complication of dis- tinct dimensional cases, as illustrated in Theorems 7.2 and 7.3, which characterizes so many results in Lorentzian warped product analysis.
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