Lorentzian Clairaut submersions
Starting from a Clairaut theorem (for geodesics on a surface of revolution), \textit{R. L. Bishop} [Differ. Geom. in honor of K. Yano, 21-31 (1972; Zbl 0246.53048)] studied Riemannian submersions \(\pi:M\to B\) for which there is a function \(r:M\to\mathbb{R}^+\) such that for every geodesic making angles \(\omega\) with the horizontal subspaces, the term \(r\sin\omega\) is constant. The author studies the above idea in the case of Lorentzian submersions. Suppose \(\gamma\) is a timelike geodesic. Then \(\gamma'=E=X+V\), where \(X\) is horizontal and \(V\) is vertical. At each point \(\gamma(s)\) the author defines \(\varphi(s)\) to be the hyperbolic angle between \(E\) and \(V\). Definition. A Lorentzian submersion \(\pi:M\to B\) is said to be a Clairaut submersion if there is a positive function (called girth) \(r:M\to\mathbb{R}\) such that for every timelike geodesic \(\gamma\) in \(M\), the term \(r\text{ cosh }\varphi\) is constant along \(\gamma\). Theorem. A Lorentzian submersion with one-dimensional fibers is a Clairaut submersion with girth \(r=\exp f\) if and only if the fibers are totally umbilic with a gradient field \(N=-\text{grad }f\) as the normal curvature vector field. Furthermore, for a Clairaut submersion having connected fibers, \(f=f_*\circ\pi\) for some \(f_*: B\to\mathbb{R}\) and \(N\) is the base vector field obtained by lifting \(N_*=-\text{grad }f_*\) horizontally. Let \(F\) be the manifold \(((a,b),-dt^2)\), \(-\infty\leq a<b\leq\infty\). From above theorem it follows that the warped product \(M=F_r\times B\) is a Clairaut submersion with girth \(r\). This implies that examples of Lorentzian submersions include universal anti-de Sitter spacetime and the external Schwarzschild spacetime. Then the author proves the theorem, from which it follows that a Clairaut submersion with integrability tensor \(A=0\) is locally a Lorentzian warped product \(F_r\times B\). In the last section the geodesics in a Lorentzian submersion are studied. In particular, it is proved that every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic (i.e., a curve which can be reparametrized to be a geodesic) in the base space with respect to a conformally related metric on the base if and only if the integrability tensor of the submersion vanishes.
scientific article; zbMATH DE number 955329