Abstract
In 1962, Ehlers and Kundt conjectured that plane waves are the only class of complete Ricci-flat pp-waves, i.e. metrics on R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>4$$\end{document} of the form ds2=2dudv+dx2+dy2+H(x,y,u)du2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ds<^>2=2du\,dv+dx<^>2+dy<^>2+H(x,y,u)du<^>2\,. $$\end{document}Recently, Flores and S & aacute;nchez gave a proof of the conjecture in the fundamental case of spatially polynomially bounded profile functions H. However, impulsivepp-waves, i.e. waves with concentrated profile functions of the form H(x,y,u)=f(x,y)delta(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x,y,u)=f(x,y)\,\delta (u)$$\end{document} (delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}, the Dirac measure) have been found to be complete for arbitrary (smooth) spatial profile functions f. We summarise completeness results for several classes of impulsive wave spacetimes achieved during the last years and discuss them in the context of the Ehlers-Kundt conjecture.