On completeness of certain families of semi-Riemannian manifolds (Q1340215)

This paper begins (section 1) by giving a brief résumé of results concerning timelike, spacelike, null and general (geodesic) completeness in manifolds with a metric of arbitrary signature. The main new results are confined to sections 2 and 3 of the paper. In section 2 the completeness of a manifold is related to the existence of certain conformal vector fields. The authors prove that if \((M,g)\) is a manifold with indefinite metric \(g\) of index \(s\) and if \(\xi_ 1\dots \xi_ s\) are timelike conformal vector fields satisfying: (i) if \(g^{ij}\) is the inverse of the matrix \(g(\zeta_ i \zeta_ j)\) then \(\Sigma (g^{ij})^ 2\) is bounded on \(M\); (ii) the functions \(\sigma_ I\) defined by \({\mathcal L}_{\xi_ i} g= \sigma_ i g\) are bounded on \(M\); (iii) the associated positive definite metric \(\gamma\) defined by \(\gamma(X,Y)= g(X,Y)\), \(\gamma(A, X)=0\), \(\gamma(A,B)=- g(A,B)\) for any \(A,B\in \text{span} \{\zeta_ 1\dots \zeta_ s\}\) and \(X\), \(Y\) in its \(g\)-orthogonal complement, is complete. Then \((M,g)\) is complete. In section 3 the authors consider warped products of manifolds. They prove that if \((B,g)\) is a manifold with indefinite metric \(g\) and if \(f: B\to \mathbb{R}\), \(f>0\), then the following is equivalent: (i) There exists a complete manifold \((N_ 0, \gamma_ 0)\) with \(\gamma_ 0\) indefinite and a fibre bundle \((E_ 0 (B, N_ 0), g^ f)\) is complete; (ii) \((E(B, F)g^ f)\) is complete for every complete manifold \((F, \gamma)\) with \(\gamma\) indefinite. If \(g\) is positive definite a modified condition (i) is given. Also, in the above, ``complete'' can be replaced by either timelike, null, or spacelike complete. The authors finish the paper with a discussion of twisted products.