Manifolds admitting a metric with co-index of symmetry 4

Abstract

By a recent result, it is known that compact homogeneous spaces with co-index of symmetry 4 are quotients of a semisimple Lie group of dimension at most 10. In this paper we determine exactly which ones of these spaces actually admit such a metric. For all the admissible spaces we construct explicit examples of these metrics.

Strongly symmetric autoparallel distributions

We introduce shortly the concept of strongly symmetric distribution. The full reference for this “Appendix” is the article [2]. Let $M = G/H$ be a compact homogeneous Riemannian manifold. Assume that G is connected and its action on M is effective. We say that a G-invariant autoparallel distribution $\mathcal D$ is strongly symmetric with respect to G if every integral manifold L(p) is a globally symmetric space and for each $v \in \mathcal D_p$ there exists a Killing field X on M, which is induced by G, such that $X_p = v$ and $X|_{L(p)}$ is parallel at p.

Example A.1

The distribution of symmetry of M is strongly symmetric with respect to the full isometry group.

Example A.2

([2, Lemma 3.11]) If $\mathcal D$ is strongly symmetric with respect to G and $\mathcal D'$ is a G-invariant autoparallel distribution such that $\mathcal D \subset \mathcal D'$ and ${{\,\mathrm{rank}\,}}\mathcal D' - {{\,\mathrm{rank}\,}}\mathcal D = 1$, then $\mathcal D'$ is strongly symmetric with respect to G.

Theorem 2.2 has a weaker version for strongly symmetric distributions.

Theorem A.3

([2, Theorem 3.7]) Let $\mathcal {D}$ be strongly symmetric with respect to G and let $k = \dim M - {{\,\mathrm{rank}\,}}{\mathcal {D}}$. Assume that M does not have a symmetric de Rham factor with associated parallel distribution contained in D. Then there exists a transitive semisimple Lie group $G' \subset G$ such that $\dim G' \le k(k + 1)/2$. Moreover, equality holds if and only the Lie algebra of $G'$ is isomorphic to $\mathfrak {so}(k + 1)$.