On the structure group of a decomposable model space

TL;DR

This paper investigates the structure group of algebraic curvature tensors derived from symmetric bilinear forms, revealing how it permutes subspaces and relates to isometry groups, with applications to manifold invariants.

Contribution

It characterizes the structure group of decomposable algebraic curvature tensors, showing its relation to wreath products of pseudo-orthogonal groups and symmetric groups, and applies findings to manifold invariants.

Findings

Structure group permutes subspaces in decomposable models.

Invariance of subspaces depends on dimension and signature.

Structure group often isomorphic to wreath products of pseudo-orthogonal groups.

Abstract

We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature tensors on a decomposable model space must permute the subspaces $V_i$ on which they are defined. For such an algebraic curvature tensor, we show that if the vector space $V$ is a direct sum of subspaces $V_1$ and $V_2$, the corresponding structure group decomposes as well if $V_1$ and $V_2$ are invariant of the action of the structure group on $V$. We determine the freedom one has in permuting these subspaces, and show these subspaces are invariant if $\dim V_1 \neq \dim V_2$ or if the corresponding symmetric forms defined on those subspaces have different (but not reversed) signatures, so that in this situation, only the trivial permutation is allowable. We exhibit a model space that realizes the full permutation group, and, with exception to the balanced signature case, show the corresponding structure group is isomorphic to the wreath product of the structure group of a given symmetric bilinear form by the symmetric group. Using these results, we conclude that the structure group of any member of this family is isomorphic to a direct product of wreath products of pseudo-orthogonal groups by certain subgroups of the symmetric group. Finally, we apply our results to two families of manifolds to generate new isometry invariants that are not of Weyl type.