Local rigidity and nonrigidity of symplectic pairs

Abstract

It is shown that indefinite strictly almost Kähler and opposite Kähler structures (J, J ) on a four-dimensional manifold with J -invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drȃghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraKähler and opposite paraKähler structures (J, J ) with J-invariant Ricci operator are shown.

Geometric properties of generalized symmetric spaces

“…Illustrating the previous result, recall from[18] the following simple examples corresponding to the different possibilities in lemma 4.4.…”

“…Riemannian fourdimensional manifolds admitting a strictly almost Kähler structure (g, J + ) and opposite Kähler structure (g, J − ) such that the Ricci operator is J + -invariant have been investigated in [3] showing that they are homogeneous and locally isometric to the unique Riemannian generalized symmetric space (see also [29]). This result has been extended by the authors in [17] to the neutral signature case, thus resulting in the following characterization of Type I generalized symmetric spaces. (Recall that the roles of J + and J − must be changed in the following theorem when considering the neutral signature case, since J ± and Ω ± induce opposite orientations in that setting).…”

“…Moreover, even in the homogeneous case, the authors showed in [18] the existence of strictly almost para-Kähler structures and opposite para-Kähler structures with invariant Ricci operator that are not locally isometric to any type II generalized symmetric spaces. A detailed examination of four-dimensional Lie groups admitting a pair of structures as above leads to the following possibilities.…”

“…[3,17] Let (M, g) be a four-dimensional pseudo-Riemannian manifold. Then (M, g) is locally isometric to a Type I generalized symmetric space if and only if (M, g) admits a strictly almost Kähler structure (g, J + ) and opposite Kähler structure (g, J − ) such that the Ricci operator is J + -invariant.Remark 10.…”