LIFTS OF GOLDEN STRUCTURES ON THE TANGENT BUNDLE

The present paper aims to study the complete lift of golden structure on tangent bundles. Integrability conditions for complete lift and third-order tangent bundle are established.

N = 4 near-horizon geometries in D = 11 supergravity

Extreme near-horizon geometries in D = 11 supergravity preserving four supersymmetries are classified. It is shown that the Killing spinors fall into three possible orbits, corresponding to pairs of spinors defined on the spatial cross-sections of the horizon which have isotropy groups SU(3), G2, or SU(4). In each case, the conditions on the geometry and the 4-form flux are determined. The integrability conditions obtained from the Killing spinor equations are also investigated.

On a class of integrable Hamiltonian equations in 2+1 dimensions

We classify integrable Hamiltonian equations of the form u t = ∂ x ( δ H δ u ) , H = ∫ h ( u , w ) d x d y , where the Hamiltonian density h ( u , w ) is a function of two variables: dependent variable u and the non-locality w = ∂ x − 1 ∂ y u . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h ). We show that the generic integrable density is expressed in terms of the Weierstrass σ -function: h ( u , w ) = σ ( u ) e w . Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Second-order integrable Lagrangians and WDVV equations

We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

Radical transversal SCR-lightlike submanifolds of indefinite Sasakian manifolds

In this paper, we introduce the notion of radical transversal screen Cauchy-Riemann (SCR)- lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some nontrivial examples of such submanifolds. Integrability conditions of distributions D1, D2, D and D? on radical transversal SCR-lightlike submanifolds of an indefinite Sasakian manifold have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

ON CAUCHY-RIEMANN STRUCTURES AND THE GENERAL EVEN ORDER STRUCTURE

The present paper aims to study the Cauchy-Riemann structures and the general even order structure and find the general even order structure that acts on complementary distributions and as an almost complex structure and a null operator, respectively. We also discuss integrability conditions and prove certain theorems on the Cauchy-Riemann structures and the general even order structure. Moreover, we construct examples of it.

SHORTENED MAPPINGS OF SPACES WITH AFFINE CONNECTIVITY

The long history of theory of mappings was revived thanks to the tensor methods of inquiry. The notion of affine connectivity was introduced a hundred years ago. It enabled us to look at classic geometric problems from a different angle. Following the common tradition, this paper introduces a notion of a mapping for a space of affine connectivity. Modifying the method of A. P. Norden, we found the formulae for the main tensors: deformation tensor, Riemann tensor, Ricci tensor and their first and second covariant derivatives for spaces and , which are connected by a given mapping. These formulae contain both objects of and with covariant derivatives in respect to relevant connectivities. In order to simplify the expression, we introduced the notion of shortened mapping and its particular case: a half-mapping. The connectivity that appears in the case of a half-mapping is called a medium connectivity. The above mentioned formulae can be notably simplified in the case of transition to covariant derivatives in the medium connectivity. This fact permits us to obtain characteristics (the necessary conditions) for the estimates whether an object of inner character from the space of affine connectivity is preserved under a given type of mappings. Objects of the inner character are geometric objects implied by an affine connectivity. They include Riemann tensor, Ricci tensor, Weyl tensor. Every type of mapping received its own set of differential equations in covariant derivatives, which define a deformation tensor of connectivity with a necessity. The study of these equations can proceed by a research on integrability conditions. Integrability conditions are algebraic over-defined systems. That’s why there is a constant need in introduction of additionally specialized spaces or certain objects of these spaces. Applying the method of N. S. Sinyukov and J. Mikes, in the case of certain algebraic conditions, we obtained a form of a deformation tensor for a given mapping. Let us note that the medium connectivity was selected in order to simplify the calculations. Depending on the type of a model under consideration or on the physical limitations, we can construct any other connectivity (and mappings), which would be better suited for the given conditions. This approach is particularly fruitful when applied for invariant transformations connecting pairs of spaces of affine connectivity via their deformation tensor of connectivity.