Affine manifolds and orbits of algebraic groups

Abstract:

This paper is the sequel to The radiance obstruction and parallel forms on affine manifolds (Trans. Amer. Math. Soc. 286 (1984), 629-649) which introduced a new family of secondary characteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold ${M^n}$ to the action on ${{\mathbf {R}}^n}$ of the algebraic hull ${\mathbf {A}}(\Gamma )$ of the affine holonomy group $\Gamma \subseteq \operatorname {Aff}({{\mathbf {R}}^n})$. A main technical result of the paper is that if $M$ has a nonzero cohomology class represented by a parallel $k$-form, then every orbit of ${\mathbf {A}}(\Gamma )$ has dimension $\geq k$. When $M$ is compact, then ${\mathbf {A}}(\Gamma )$ acts transitively provided that $M$ is complete or has parallel volume; the converse holds when $\Gamma$ is nilpotent. A $4$-dimensional subgroup of $\operatorname {Aff}({{\mathbf {R}}^3})$ is exhibited which does not contain the holonomy group of any compact affine $3$-manifold. When $M$ has solvable holonomy and is complete, then $M$ must have parallel volume. Conversely, if $M$ has parallel volume and is of the homotopy type of a solvmanifold, then $M$ is complete. If $M$ is a compact homogeneous affine manifold or if $M$ possesses a rational Riemannian metric, then it is shown that the conditions of parallel volume and completeness are equivalent.