A para-Kahler manifold can be defined as a pseudo-Riemannian manifold with a parallel skew-symmetric para-complex structure , that is, a parallel field of skew-symmetric endomorphisms with or, equivalently, as a symplectic manifold with a bi-Lagrangian structure , that is, two complementary integrable Lagrangian distributions. A homogeneous manifold of a semisimple Lie group admits an invariant para-Kahler structure if and only if it is a covering of the adjoint orbit of a semisimple element . A description is given of all invariant para-Kahler structures on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kahler geometry it is proved that any invariant para-complex structure on defines a unique para-Kahler Einstein structure with given non-zero scalar curvature. An explicit formula for the Einstein metric is given. A survey of recent results on para-complex geometry is included.

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Homogeneous para-Kähler Einstein manifolds

A para-K\"ahler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $ K^2 = \mathrm{Id} $ or, equivalently, as a symplectic manifold $(M,\omega)$ with a bi-Lagrangian structure $L^\pm$, i.e. two complementary integrable Lagrangian distributions. 
A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an invariant para-K\"ahler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\mathrm{Ad}_Gh$ of a semisimple element $h.$ We give a description of all invariant para-K\"ahler structures $(g,K)$ on a such homogeneous manifold. Using a para-complex analogue of basic formulas of K\"ahler geometry, we prove that any invariant para-complex structure $K$ on $M = G/H$ defines a unique para-K\"ahler Einstein structure $(g,K)$ with given non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is given. 
A survey of recent results on para-complex geometry is included.

Homogeneous para-K\"ahler Einstein manifolds

Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups

Let ( M , g ) be a compact Riemannian manifold and T 1 M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from ( M , g ) to ( T 1 M , g ˜ s ) , g ˜ s being the Sasaki metric on T 1 M , have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T 1 M , are particular examples of g-natural metrics. We equip T 1 M with an arbitrary Riemannian g-natural metric G ˜ , and investigate the harmonicity of a unit vector field V of M, thought as a map from ( M , g ) to ( T 1 M , G ˜ ) . We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold.

Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

We define the notion of co-locally weakly differentiable maps from a
manifold M to a manifold N. If p ! 1 and if the manifolds M and N are endowed
with a Riemannian metric, this allows us to define intrinsically the homogeneous
Sobolev space W˙ 1,p(M, N). This new definition is equivalent to the definition by
embedding in the Euclidean space and to that of Sobolev map into a metric space.
The co-local weak derivative is an approximate derivative. The co-local weak
differentiability is stable under a suitable weak convergence. The Sobolev spaces
can be endowed with various intrinsic distances that induce the same topology
and for which the space is complete.

/pdf/intrinsic-colocal-weak-derivatives-and-sobolev-spaces-2welxt3au6.pdf

Intrinsic colocal weak derivatives and Sobolev spaces between manifolds

We study harmonic sections of a Riemannian vector bundle E → M when E is equipped with a 2-parameter family of metrics hp,q which includes both the Sasaki and Cheeger–Gromoll metrics. For every k> 0 there exists a unique p such that the harmonic sections of the radius-k sphere subbundle are harmonic sections of E with respect to hp,q for all q. In both compact and noncompact cases, Bernstein regions of the (p, q)-plane are identified, where the only harmonic sections of E with respect to hp,q are parallel. Examples are constructed of vector fields which are harmonic sections of E = TM in the case where M is compact and has non-zero Euler characteristic.

Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type

We study the geometry of the tangent bundle equipped with a two-parameter family of Riemannian metrics. After deriving the expression of the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. Specializing to the case of space forms, we characterise the metrics giving positive sectional curvature and show that one can always find parameters ensuring positive scalar curvature on the tangent space. Under some curvature conditions, this extends to general base manifolds.

The geometry of generalised Cheeger-Gromoll metrics

We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restrictions to the fibres have positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

https://projecteuclid.org/download/pdf_1/euclid.tjm/1264170234

The Geometry of Generalised Cheeger-Gromoll Metrics

This article studies the harmonicity of vector fields on Rie- mannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigid- ity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Fi- nally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.

/pdf/harmonic-maps-and-kaluza-klein-metrics-on-spheres-3ytzhmo0pq.pdf

Harmonic maps and Kaluza-Klein metrics on spheres

A semi-Riemannian (called also pseudo-Riemannian) manifold (M, G) [8], endowed with a product structure P (i.e. P 2 = I, P ≠ ±I), which is skew-symmetric with respect to G, is called an almost para-Hermitian manifold. These manifold have been introduced in [7],[10] and then many other authors have studied their classification, their submanifolds (which can be either degenerate or nondegenerate), their applications in theoretical physics and so on. The notion of para-Hermitian structure can be extended from the tangent bundle of a manifold to an arbitrary vector bundle, [1].

M. Benyounes

Papers

Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type

The geometry of generalised Cheeger-Gromoll metrics

The Geometry of Generalised Cheeger-Gromoll Metrics

Harmonic maps and Kaluza-Klein metrics on spheres

Harmonic Maps Between Almost Para-Hermitian Manifolds