On formality of generalized symmetric spaces

Abstract

We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

Chern numbers and the geometry of partial flag manifolds

“…The latter is a symmetric space, and therefore geometrically formal. The case n D 1 was proved in [17], where we also considered other homogeneous spaces G=H where H is a torus. The following proof shows that the arguments of [17] apply much more generally.…”

Curvature Properties of Four-Dimensional Generalized Symmetric Spaces

“…More recently, S. Terzić classified generalized symmetric spaces defined as quotients of compact simple Lie groups, describing explicitly their real cohomology algebras [17] and calculating their real Pontryagin characteristic classes [18]. Moreover, D. Kotschick and S. Terzić [13] proved that all generalized symmetric spaces are formal (their rational homotopy type is determined by their rational cohomology algebra alone), but in many cases they are not geometrically formal, that is, the product of harmonic forms is not always harmonic.…”

Rational homotopy groups of generalised symmetric spaces

“…In [25] it was proved that all generalised symmetric spaces of simple compact Lie groups are Cartan pair homogeneous spaces. Therefore [10] they are formal in the sense of Sullivan.…”