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

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A Survey on Paracomplex Geometry

We study the linear stability of compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. Furthermore, the failure of the uniform Kreiss-Lopatinskii condition yields a loss of derivatives in the energy estimate.

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The stability of compressible vortex sheets in two space dimensions

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations We also study the non-integrable case (ie, a symplectic manifold endowed with two transversal Lagrangian distributions) We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples

The geometry of a bi-Lagrangian manifold

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Nonsymmetric osserman pseudo-riemannian manifolds

One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with – = -1 is replaced by \( \tau \)with \( \tau^{2} = 1 \). A rather surprising amount of complex geometry carries over, almost untouched, and this has been the subject of many papers. We briefly review this material and, in particular, we discuss Hermitian D-manifolds with trivial canonical bundle, which provide the background space for the geometry of split SLAG submanifolds. A removable singularities result is proved for split SLAG subvarieties. It implies, in particular, that there exist no split SLAG cones, smooth outside the origin, other than planes. This is in sharp contrast to the complex case. Parallel to the complex case, space-like Lagrangian submanifolds are stationary if and only if they are Ѳ-split SLAG for some constant phase angle Ѳ, and infinitesimal deformations of split SLAG submanifolds are characterized by harmonic 1-forms on the submanifold. We also briefly review the recent work of Kim, McCann and Warren who have shown that split Special Lagrangian geometry is directly related to the Monge-Kantorovich mass transport problem.

Split Special Lagrangian Geometry

We consider the sectional curvatures for metric (J4 = 1)-manifolds, and study particularly the general expression of the metric and almostproduct structure in normal coordinates for para-Kaehlerian manifolds of constant para-holomorphic sectional curvature. We also introduce models of such spaces.

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Spaces of constant para-holomorphic sectional curvature

The Kahler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kahler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kahler manifold admits a nonvanishing Kahler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

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Characterizing the complex hyperbolic space by Kähler homogeneous structures

Partiellement supporte par la DGES (Espagne) : P.M.G. et J.M.M. sous Ie Projet PB-95-0124 et A.M.A. sous Ie Projet PB-94-0972

A. Montesinos Amilibia

Papers

Spaces of constant para-holomorphic sectional curvature

Characterizing the complex hyperbolic space by Kähler homogeneous structures

Abbena-Garbiero's class K2 ⊕ K4 of Kähler homogeneous structures | La classe K2 ⊕ K4 d'Abbena et Garbiero des structures homogènes kählériennes