Paracomplex Manifolds Research Articles

The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations

A Riemannian almost paracomplex manifold is a 2n-dimensional Riemannian manifold (M,g), whose structural group O(2n,R) is reduced to the form O(n,R)×O(n,R). We define the scalar curvature π of this manifold and consider relationships between π and the scalar curvature s of the metric g and its conformal transformations.

Paraholomorphic cohomology groups of hyperbolic adjoint orbits

For a paracomplex manifold, we construct certain cohomology groups. The main purpose of this paper is to clarify a link between such cohomology groups of hyperbolic adjoint orbits and the de Rham cohomology groups of real flag manifolds. That establishes relation between a paraholomorphic invariant and a topological invariant.

A note on para-holomorphic Riemannian–Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

Almost Analyticity on Almost (Para) Complex Lie Algebroids

The goal of this paper is to generalize the notion of almost analyticity in the almost (para)complex Lie algebroids framework. We use a global formalism which yields to generalizations of the main results of previous known almost (para)complex manifolds case.

On 4-dimensional almost para-complex pure-Walker manifolds

This paper is concerned with almost para-complex structures on Walker 4-manifolds. For these structures, we study some problems of Kähler manifolds. We also give an example of a flat almost para-complex manifold, which consists of a nonintegrable almost para-complex structure on Walker 4-manifolds.

Lie groupoids and generalized almost paracomplex manifolds

In this paper, we show that there is a close relationship between generalized paracomplex manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost paracomplex manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized paraholomorphic maps.

On small deformations of paracomplex manifolds

A paracomplex structure on a manifold M is an endomorphism K of the tangent bundle TM such that K^2= I , whose \pm 1 -eigenspaces have the same dimension and are involutive. By using the theory of differential graded Lie algebras, we describe small deformations of paracomplex structures. We also compute the space of invariant small deformations of 4-dimensional nilmanifolds endowed with a fixed paracomplex structure.

Harmonic bundle solutions of topological–antitopological fusion in para-complex geometry

In this work we introduce the notion of a para-harmonic bundle, i.e. the generalization of a harmonic bundle [C.T. Simpson, Higgs-bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992) 5–95] to para-complex differential geometry. We show that para-harmonic bundles are solutions of the para-complex version of metric t t ∗ -bundles introduced in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89]. Further we analyze the correspondence between metric para- t t ∗ -bundles of rank 2 r over a para-complex manifold M and para-pluriharmonic maps from M into the pseudo-Riemannian symmetric space GL ( r , R ) / O ( p , q ) , which was shown in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89], in the case of a para-harmonic bundle. It is proven, that for para-harmonic bundles the associated para-pluriharmonic maps take values in the totally geodesic subspace GL ( r , C ) / U π ( C r ) of GL ( 2 r , R ) / O ( r , r ) . This defines a map Φ from para-harmonic bundles over M to para-pluriharmonic maps from M to GL ( r , C ) / U π ( C r ) . The image of Φ is also characterized in the paper.

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt *-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt *-bundles with \({\nabla=D + S}\) induced by the one-parameter family of connections given by \({\nabla^{\theta}=\exp(\theta \tau) \circ \nabla \circ\exp(-\theta \tau)}\) and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-Kahler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt *-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt *-bundles generalised para-pluriharmonic maps into \({{{\rm Sp}(\mathbb{R}^{2n})/U^{\pi}(C^n)}}\) , respectively, into SO 0(n,n)/U π(C n ), where U π(C n ) is the para-complex analogue of the unitary group.

Decompositions of para-complex vector bundles and para-complex affine immersions

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ∇ over a para-complex manifold. First we obtain results on the connections induced on the subbundles, their second fundamental forms and their curvature tensors. In particular we analyze para-holomorphic decompositions. Then we introduce the notion of para-complex affine immersions and apply the above results to obtain existence and uniqueness theorems for para-complex affine immersions. This is a generalization of the results obtained by Abe and Kurosu [AK] to para-complex geometry. Further we prove that any connection with vanishing (0, 2)-curvature, with respect to the grading defined by the para-complex structure, induces a unique para-holomorphic structure.

Formula omitted]-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps

We introduce the notion of a para- t t ∗ -bundle, the generalization of a t t ∗ -bundle (compare [V. Cortés, L. Schäfer, Topological–antitopological fusion equations, pluriharmonic maps and special Kähler manifolds, in: O. Kowalski, E. Musso, D. Perrone (Eds.), Proceedings of the Conference “Curvature in Geometry” organized in Lecce in honor of Lieven Vanhecke, Progress in Mathematics, vol. 234, Birkhäuser, 2005] and [L. Schäfer, t t ∗ -geometry and pluriharmonic maps, Ann. Global Anal. Geom., in press]) in para-complex geometry. The main result is the definition of a map Φ from the space of metric para- t t ∗ -bundles of rank r over a para-complex manifold M to the space of para-pluriharmonic maps from M to GL ( r ) / O ( p , q ) where ( p , q ) is the signature of the metric and the description of the image of this map Φ. Then we recall and prove some results known in special complex and special Kähler geometry in the setting of para-complex geometry, which we use in the sequel to give a simple characterization of the tangent bundle of a special para-complex and special para-Kähler manifold as a particular type of t t ∗ -bundles. For the case of a special para-Kähler manifold it is shown that the para-pluriharmonic map coincides with the dual Gauß map, which is a para-holomorphic map into the symmetric space Sp ( R 2 n ) / U π ( C n ) ⊂ SL ( 2 n ) / SO ( n , n ) ⊂ GL ( 2 n ) / O ( n , n ) .

Dirac structures and paracomplex manifolds

We introduce the notion of a generalized paracomplex structure. This is a natural notion which unifies several geometric structures such as symplectic forms, paracomplex structures, and Poisson structures. We show that generalized paracomplex structures are in one-to-one correspondence with pairs of transversal Dirac structures on a smooth manifold. To cite this article: A. Wade, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kahler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special Kahler manifolds appearing in Minkowskian N=2 supersymmetry is discussed. Starting from the general 5-dimensional vector multiplet action we consider dimensional reduction over time and space in parallel, providing a dictionary between the resulting Euclidean and Minkowskian theories. Then we reanalyze supersymmetry in four dimensions and find that any (para-)holomorphic prepotential defines a supersymmetric Lagrangian, provided that we add a specific four-fermion term, which cannot be obtained by dimensional reduction. We show that the Euclidean action and supersymmetry transformations, when written in terms of para-holomorphic coordinates, take exactly the same form as their Minkowskian counterparts. The appearance of a para-complex and complex structure in the Euclidean and Minkowskian theory, respectively, is traced back to properties of the underlying R-symmetry groups. Finally, we indicate how our work will be extended to other types of multiplets and to supergravity in the future and explain the relevance of this project for the study of instantons, solitons and cosmological solutions in supergravity and M-theory.

Self-duality in 3 + 3 dimensions and the KP equation

We consider two types of generalized self-duality conditions for Yang-Mills fields on paracomplex manifolds of arbitrary dimension. We then specialize to 3 + 3 dimensions and show how one can obtain the KP equation from these self-duality conditions on SL (2, R ) valued gauge fields.

Almost paracontact and parahodge structures on manifolds

In this paper we study the paracomplex analogues of almost contact structures, and we introduce and study the notion of parahodge structures on manifolds. In particular, we construct new examples of paracomplex manifolds and we find all simply connected parahermitian symmetric coset spaces, which are the adjoint orbits of noncompact simple Lie groups, with parahodge structures induced by the Killing forms. This is done by (i) observing that a version of the results of A. Morimoto [4] on almost contact structures can be formulated and proved for almost paracontact structures, and by (ii) the methods of geometric quantization [3] applied to parahermitian symmetric triples [1] in conjunction with results of [7]. Two of the main results are Theorem 2.5 (which ties together the above structures) and Corollary 3.9.