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Sectional curvature, symmetries, and conformally flat plane waves. (English) Zbl 0758.53052
The sectional curvature determines the metric of a (4-dimensional) spacetime, unless spacetime is a conformally flat generalized pp wave, i.e. admits coordinates $(x,y,u,v)$ such that $$g=dx\sp 2+dy\sp 2+2dudv+{1\over 2}A(u)(x\sp 2+y\sp 2)du\sp 2$$ [{\it G. S. Hall}, Gen. Relativ. Gravitation 16, 79-88 (1984; Zbl 0535.53023)]. In the present article the authors study vector fields in conformally flat generalized pp waves which leave all sectional curvatures invariant. They show that these vector fields constitute a Lie algebra of dimension 6, 7 or 8. (The (sub) Lie algebra of Killing vector fields has dimension 6 or 7.) They calculate the structure of the Lie algebra and give a procedure to find conformally flat generalized pp waves for each possible dimension combination $((6,6),\ (7,6),\ (8,6),\ (7,7))$.

53Z05Applications of differential geometry to physics
83C35Gravitational waves
83C20Classes of solutions of equations in general relativity
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