Dependencies of bundle forms in Laguerre planes

Abstract

Several relationships between bundle forms of Laguerre planes are found, including the results (1) \({B\mathfrak{B} 0}\), \({D\mathfrak{B}0}\), \({B\mathfrak{B}1^1}\), \({B\mathfrak{B}1^2}\), \({D\mathfrak{B}1}\), \({BD\mathfrak{B}1_{2}}\), \({DD\mathfrak{B}1}\) are equivalent, and (2) \({BD\mathfrak{B}0}\), \({BBD\mathfrak{B}1}\), \({BDD\mathfrak{B}1^1}\), \({BDD\mathfrak{B}1^2}\) are equivalent. A graph indicating all known dependencies between bundle forms in Laguerre planes is provided. It is shown that \({BD\mathfrak{B}2}\) holds in any \({\mathcal{G}}\)-translation Laguerre plane, that is, any Laguerre plane that is a G-translation plane for each parallel class G of spears. A simplified graph of dependencies is supplied for these planes.