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Incompleteness of timelike submanifolds with nonvanishing second fundamental form. (English) Zbl 0566.53057

Let f :M\(\to L^ n\) be an immersion of M as a timelike hypersurface of \(L^ n\). If the inverse image of each compact set in \(L^ n\) is compact, then M is properly immersed. If N is a unit spacelike normal to M, then the second fundamental form \(S_ N: T_ pM\times T_ pM\to {\mathbb{R}}\) need not be diagonalizable. If it is diagonalizable and positive definite on \(T_ pM\), then dim \(M\geq 3\) implies Ric (v,v)\(>0\) for all nonzero nonspacelike vectors. It follows that under these assumptions M satisfies both the generic and strong energy conditions. If M is also causally disconnected, then M is nonspacelike geodesically incomplete.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C40 Global submanifolds
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