The present paper will be referred to as Part I. A subsequent paper entitled, “Lie algebra cohomology and generalized Schubert cells,” will be referred to as Part II.

Sur une classe remarquable d'espaces de Riemann

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

Les sous-groupes fermés de rang maximum des groupes de Lie clos.

We shall call paracomplex geometry the geometry related to the algebra of paracomplex numbers [70] and, mainly, the study of the structures on differentiable manifolds called paracomplex structures. When we moreover consider a compatible neutral pseudo-Riemannian metric, we have the paraHermitian and para-Kähler structures, and their variants. This subject has been studied, since the first papers by Rashevskij [94], Libermann [69] and Patterson [90] until now, from several different points of view. Moreover, the papers related to it have appeared many times in a rather disperse way, and the different schools or authors have worked many times having no relation with one another. So, the interest for a survey on the topic is clear; and, of course, mathematicians, and in particular differential geometers, will find a lot of definitions, examples, results and references in paracomplex geometry. Furthermore, we think that the solved problems in paracomplex geometry are only a little part of the “first” questions to be studied in the theory. And this is one of the attractive features of it: to be de facto a young branch of Differential Geometry, with many significant results to be proved, as the reading of the present survey, we hope, shows. Nowadays, the subject has already applications to several topics, as: negatively curved manifolds and Anosov diffeomorphisms [47], quantizable coadjoint orbits [57], mechanics [11, 86], chronogeometry [82], elliptic geometry [32], pseudo-Riemannian space forms [38], etc. On the other hand, as is seen in certain recent works, paracomplex geometry is related to some physical problems, and we think that this work can also be useful for physicists and mathematicians working in them. In particular, since every almost para-Kählerian manifold is symplectic, those manifolds furnish a large family of symplectic manifolds. Moreover, as Kaneyuki [49] points out, the study of global geometric properties of affine symmetric spaces seems to be an interesting and important problem, since affine symmetric spaces have begun to play a central role in the theory of group representations, as is shown in [13, 30, 87]. In

background

/pdf/a-survey-on-paracomplex-geometry-2f6cqov0cb.pdf

A Survey on Paracomplex Geometry

On Compact, Locally Symmetric Kahler Manifolds

/pdf/on-root-systems-and-an-infinitesimal-classification-of-1zdobkfat8.pdf

On root systems and an infinitesimal classification of irreducible symmetric spaces

/pdf/sur-une-classe-remarquable-d-espaces-de-riemann-586t7490e1.pdf

Sur une classe remarquable d'espaces de Riemann

Citations

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

Les sous-groupes fermés de rang maximum des groupes de Lie clos.

A Survey on Paracomplex Geometry

On Compact, Locally Symmetric Kahler Manifolds

On root systems and an infinitesimal classification of irreducible symmetric spaces

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