Mutual exclusiveness among spacelike, timelike, and lightlike leaves in totally geodesic foliations of lightlike complete Lorentzian two-dimensional tori
Let \((M,g)\) be a Lorentzian manifold and \(T_xM\) the tangent space at \(x\in M\). A subspace \(E\subset T_xM\) is called spacelike (resp. timelike, lightlike) if the signature of \(g/E\) is \((+,\dots,+)\) (resp. \((+, \dots,+,-)\), \((+,\dots, +,0))\). A submanifold \(L\) of \((M,g)\) is called spacelike (resp. timelike, lightlike) if the tangent space \(T_xL\) is a spacelike (resp. timelike, lightlike) subspace of \(T_xM\) for each \(x\in L\). In this paper the author proves the following Theorem: ``Let \((T^2,g)\) be a lightlike complete Lorentzian 2-torus. There exists no totally geodesic foliation which contains more than or equal to two kinds of leaves among spacelike, timelike, and lightlike ones.'' Corollary: ``Let \((T^2,g)\) be a lightlike complete Lorentzian 2-torus. Let \(F\) be a totally geodesic foliation of \(T^2\). Then \(F\) has no Reeb components.'' In addition this work covers some properties of totally geodesic foliations of Lorentzian manifolds. Some important examples are studied, too.
scientific article; zbMATH DE number 1876753