Nonexistence of invariant rigid structures and invariant almost rigid structures

TL;DR

This paper proves that certain volume-preserving Lie group actions do not preserve rigid geometric structures but do preserve invariant almost rigid structures, highlighting limitations of geometric structure gluing.

Contribution

It introduces the concept of almost rigid structures and demonstrates their invariance under specific exotic group actions, contrasting with the nonexistence of invariant rigid structures.

Findings

Certain exotic actions do not preserve rigid structures.

These actions do preserve invariant almost rigid structures.

Rigid structures cannot be extended to the entire manifold in these cases.

Abstract

We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spaces. The actions on homogeneous spaces all preserve affine connections, whereas the action along the exceptional divisor preserves a projective structure. The fact that these structures cannot in some way be "glued together" to give a rigid structure on the entire space is not obvious. We also define the notion of an almost rigid structure. The paradigmatic example of a rigid structure is a global framing and the paradigmatic example of an almost rigid structure is a framing that is degenerate along some exceptional divisor. We show that the actions discussed above do possess an invariant almost rigid structure. Gromov has shown that a manifold with rigid geometric structure invariant under a topologically transitive group action is homogeneous on an open dense set. How generally this open dense set can be taken to be the entire manifold is an important question with many dynamical applications. Our results indicate one way in which the geometric structure cannot degenerate off the open dense set.