Narrow Results

We have examined brane world solutions for the low energy effective string gravity action coupled with a quadratic dilaton potential in string frame. For a negative bulk cosmological constant, static 3-brane solutions exist. Their properties are analyzed depending on the quadratic potential parameters. It is found that a brane solution in a curved bulk space in Einstein frame corresponds to a dilaton solution in flat space in string frame.

We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists.

Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration.

We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces.

Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

More than thirty years passed since the first discoveries of various aspects of integrability of the symmetry reduced vacuum Einstein equations and electrovacuum Einstein - Maxwell equations were made and gave rise to constructions of powerful solution generating methods for these equations. In the subsequent papers, the inverse scattering approach and soliton generating techniques, Bäcklund and symmetry transformations, formulations of auxiliary Riemann-Hilbert or homogeneous Hilbert problems and various linear integral equation methods have been developed in detail and found different interesting applications. Recently many efforts of different authors were aimed at finding of generalizations of these solution generating methods to various (symmetry reduced) gravity, string gravity and supergravity models in four and higher dimensions. However, in some cases it occurred that even after the integrability of a system was evidenced, some difficulties arise which do not allow the authors to develop some effective methods for constructing of solutions. The present survey includes some remarks concerning the history of discoveries of some of the well known solution generating methods, brief descriptions of various approaches and their scopes as well as some comments concerning the possible difficulties of generalizations of various approaches to more complicate (symmetry reduced) gravity models and possible ways for avoiding these difficulties.

Gauss-Bonnet gravity provides one of the most promising frameworks to study curvature corrections to the Einstein action in supersymmetric string theories, while avoiding ghosts and keeping second order field equations. Although Schwarzschild-type solutions for Gauss-Bonnet black holes have been known for long, the Kerr-Gauss-Bonnet metric was missing. Gauss-Bonnet solutions in N = 5 dimensional space-time are discussed for spinning black holes and the related thermodynamical properties are briefly outlined.

The monodromy transform approach, developed originally for solution of integrable reductions of vacuum Einstein equations and electrovacuum Einstein - Maxwell equations in General Relativity, was shown to be applicable to solution of the field equations which govern the bosonic dynamics of string gravity in four and higher dimensions and 5D minimal supergravity for space-times with the Abelian isometry group of codimension 2. In this short communication, we discuss a choice of (matrix-valued for these cases) monodromy data for construction of solutions which satisfy physically reasonable conditions (e.g., regularity of the axis of symmetry). We describe also a convenient “canonical” form of the matrix monodromy data and some discrete non-gauge symmetries of the spectral problem which can be used to restore the generic data from these “canonical” ones.