Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences

The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems. We discuss periodic Laplace sequences of with the emphasize on the simplest nontrivial case of period 2. For 3-component systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn to be related to a special quadratic reduction of the (2+1)-dimensional 3-wave system which can be reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2+1)-dimensional n-wave system. The 2-component Laplace transformations can be interpreted also as the symmetries of the (2+1)-dimensional integrable equations of Davey-Stewartson type. Laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.Comment: 22 pages, Late

On Rank Problems for Planar Webs and Projective Structures

We present old and recent results on rank problems and linearizability of geodesic planar webs.Comment: 31 pages; LaTeX; corrected the abstract and Introduction; added reference

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.Comment: 38 pages, 8 figures, late

Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries

We briefly review the hierarchy for the hyper-K\"ahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a `twistor group' action as introduced by Bielawski \cite{B00}. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalised Legendre transform for the hyper-K\"ahler metric itself given by Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the 'finite gap' solutions in soliton theory. Finally we extend the concept of a hierarchy from that of \cite{DM00} for the four-dimensional hyper-K\"ahler equations to a generalisation of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations.Comment: Final version. To appear in the August 2003 special issue of JMP on `Integrability, Topological Solitons, and Beyond

Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem for the natural diagonal (almost) para-K\"ahlerian structures on the total spaces of the cotangent bundle.Comment: 10 pages, will appear in Czechoslovak Mathematical Journa

Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces

We use some results from polarity theory to recast several geometric properties of Conjugate Gradient-based methods, for the solution of nonsingular symmetric linear systems. This approach allows us to pursue three main theoretical objectives. First, we can provide a novel geometric perspective on the generation of conjugate directions, in the context of positive definite systems. Second, we can extend the above geometric perspective to treat the generation of conjugate directions for handling indefinite linear systems. Third, by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradient- based methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient

Invariante Konstruktion der Geometrie einer Hyperfläche in einem konformen Raum

A partly expository paper on conformal differential geometry of a hypersurface. The first three sections are introductory and are devoted to multispherical coordinate representation and the study of the subgroups of the conformal group leaving the surface element invariant. In the next four sections is given an invariant construction of the theory by means of the stationary subgroups associated with the hypersurface. By this means it is possible to introduce conformal tensors gij, aij, cij and Bk = bijkaij which determine the hypersurface within a conformal mapping. The remainder of the paper is devoted to some special problems dealing with the order of contact of certain cyclides with the hypersurface, particularly for the critical cases of the envelopping space being of 3 and 4 dimensions