EQUIVARIANT EXPONENTIALLY NASH VECTOR BUNDLES

Abstract

Let $G$ be a compact affine exponentially Nash group and let $\eta$ be a $C^\infty G$ vector bundle over a compact affine exponentially Nash $G$ manifold $X$. We prove that $\eta$ admits a unique strongly exponentially Nash $G$ vector bundle structure $\zeta$, and that $\eta$ admits a non-strongly exponentially Nash $G$ vector bundle structure if dim $X\geq 1$, rank $\eta\geq 1$ and $X$ has a 0-dimensional orbit. Moreover we show that every exponentially Nash $G$ vector bundle structure of $\eta$ which is not necessarily strongly exponentially Nash is exponentially Nash $G$ vector bundle isomorphic to $\zeta$ if the action on $X$ is transitive.