For a homogeneous space G/P , where P is a parabolic subgroup of a complex semisimple group G, an explicit Kähler–Einstein metric on it is constructed. The Einstein constant for the metric is 1. Therefore, the intersection number of the first Chern class of the holomorphic tangent bundle of G/P coincides with the volume of G/P with respect to this Kähler–Einstein metric, thus enabling us to com...

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a fami...

In this paper, we show that the analogue of Thurston's asymmetric metric on Teichmüller space flat structures torus is weak Finsler and give a geometric description its unit circle at each point in tangent to space. We then introduce family metrics which interpolate between (which coincides with hyperbolic metric). describe circles family.

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable ...

In this note, we prove the Morse index theorem for a geodesic connecting two submanifolds in $$C^7$$ manifold with $$C^6$$ (conic) pseudo-Finsler metric provided that fundamental tensor is positive definite along velocity curve of geodesic.

We investigate the isometry extension property for Einstein metrics on manifolds with boundary; namely when Killing fields of the boundary metric extend to Killing fields of any filling Einstein metric. Applications to Bartnik’s static extension conjecture are also discussed.

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the sa...

In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi-and indefinite-inner-products and define the semi-indefinite-inner-product and the corresponding structure, the semi-indefinite-inn...