Let $(P,X,G)$ be a principal bundle where $G$ is a Lie group which acts on $P$. We fix a principal connection on $P$ and a right invariant metric for $G$. These structures define a unique Riemannian metric on $P$ in an obvious way.
A $1$ dimensional subbundle $\ell$ of $TP$ is called a bisector if all its vectors have the same distance from horizontal and vertical spaces. In the other word, for every $x\in P$ and all $W_x\in \ell_x$, we have $d(W_x, H_x)=d(W_x,V_x)$ where $H_x, V_x$ are horizontal and vertical spaces, respectively. The distance $d$ arise from the inner product of $T_xP$.
Are all bisector line bundles isomorphic to each other?