# A bisector line bundle over the total space of a principal bundle

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?