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Abstract
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Introduction
Definitions and First Results
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Pure Mathematics

A Survey on Paracomplex Geometry

Profile image of Pedro AyusoPedro Ayuso

Abstract

para-Hermitian geometry, a type of affine symmetric spaces, called para-Hermitian symmetric spaces, arises (see Section 5). These spaces are useful, among other things, in the representation theory of the associated groups. Moreover, those symmetric spaces M are diffeomorphic to (a covering manifold of) the cotangent bundle of another-Riemannian-symmetric space M 0. That permits us to bring into T * (M 0) the para-Kähler structure on M , thus obtaining an additional structure on the cotangent bundle of those manifolds, which is of great interest in symplectic geometry and mechanics. Paracomplex geometry is a topic with many analogies and also with differences with complex geometry. Thus this subject is often studied with the geometries arising from complex numbers, and also with the geometry coming from dual numbers (see for instance [21, 26, 70, 109, 118]). However, for the sake of brevity, we shall make, in this survey, no references to related results on other algebras. On the other hand, a little knowledge of paracomplex geometry shows soon that there are more differences with complex geometry than one can suspect at the beginning. Perhaps, the subject would we worthy to appear in the corresponding section of the A.M.S. Subject Classification. We have entitled the present work "A survey on paracomplex geometry", but it is perhaps premature to definitively name the topic, since there are several different names, each of which having a reasonable sense; and mainly because the growth of the subject can help to fix the most reasonable name. 2 Paracomplex manifolds 2.1 Some definitions and results We shall first recall some general definitions concerning (almost) paracomplex, (almost) para-Hermitian and (almost) para-Kähler manifolds. From now on, all the manifolds and geometric objects are supposed to be C ∞. Definition 2.1 An almost product structure J on a differentiable manifold M is a (1, 1) tensor field J on M such that J 2 = 1. The pair (M, J) is called an almost product manifold. An almost paracomplex manifold is an almost product manifold (M, J) such that the two eigenbundles T + M and T − M associated with the two eigenvalues +1 and −1 of J, respectively, have the same rank. (Note that the dimension of an almost paracomplex manifold is necessarily even.) Equivalently, a splitting of the tangent bundle T M

Figures (1)
Here, for an almost para-Hermitian manifold (MM, g, J), V denotes the Levi- Civita connection, F the fundamental 2-form, 6 the codifferential, V (for vertical) the (+1)-eigendistribution associated with the eigenvalue +1 of J, H (for horizontal) the (—1)-eigendistribution corresponding to the eigenvalue —1 of J, A, B vector fields of V and U,V vector fields of H. metric g. Thus we have an adequate framework for the several types of almost Hermitian manifolds, previously defined by a number of authors in terms of geometric properties which retain some portion of Kahler geome- try. A classification of almost para-Hermitian manifolds is made in [8]. The author obtains 36 classes up to duality, and gives characterizations of some of the classes. A classification 4 la Gray-Hervella is given in [37], where 136 classes up to duality are obtained. We give here the table of primitive classes W,,..., Wg obtained in [37]:
Here, for an almost para-Hermitian manifold (MM, g, J), V denotes the Levi- Civita connection, F the fundamental 2-form, 6 the codifferential, V (for vertical) the (+1)-eigendistribution associated with the eigenvalue +1 of J, H (for horizontal) the (—1)-eigendistribution corresponding to the eigenvalue —1 of J, A, B vector fields of V and U,V vector fields of H. metric g. Thus we have an adequate framework for the several types of almost Hermitian manifolds, previously defined by a number of authors in terms of geometric properties which retain some portion of Kahler geome- try. A classification of almost para-Hermitian manifolds is made in [8]. The author obtains 36 classes up to duality, and gives characterizations of some of the classes. A classification 4 la Gray-Hervella is given in [37], where 136 classes up to duality are obtained. We give here the table of primitive classes W,,..., Wg obtained in [37]:

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