The Lifting Problem for Affine Structures in Nilpotent Lie Groups

Abstract

Affine manifolds occur in several situations in pure and applied mathematics, (e.g. leaves of Lagrangian foliations, completely integrable Hamiltonian systems, linear representations of virtually polycyclic groups, geometric quantization and so on). This work is devoted to left invariant affinely flat structures in Lie groups. We are mainly concerned with the following situation. Let $G$ and ${G_0}$ be nilpotent Lie groups of dimension $n + 1$ and $n$ , respectively and let $h:G \to {G_0}$ be a continuous homomorphism from $G$ onto ${G_0}$ . Given a left invariant affinely flat structure $({G_0},{\nabla _0})$ the lifting problem is to discover whether $G$ has a left invariant affinely flat structure $(G,\nabla )$ such that $h$ becomes an affine morphism. In the present work we answer positively when $({G_0},{\nabla _0})$ is "normal". Therefore the existence problem for a left invariant complete affinely flat structure in nilpotent Lie groups is solved by applying the following subsequent results. Let $\mathfrak {A}f({G_0})$ be the set of left invariant affinely flat structures in the nilpotent Lie group ${G_0},({1^ \circ })\;\mathfrak {A}f({G_0}) \ne \emptyset$ implies the existence of normal structure $({G_0},{\nabla _0}) \in \mathfrak {A}f({G_0});({2^ \circ })\;h:G \to {G_0}$ being as above every normal structure $({G_0},{\nabla _0})$ has a normal lifted in $\mathfrak {A}f(G)$.