In this talk I will discuss Hasimoto transformations of wave map equations and mKdV map equations using moving frames connected with integrable flows of curves in constant curvature Riemannian manifolds and semisimple Lie group manifolds. Such type of transformations have played a central role in recent analytical work on Schrodinger map equations.
In this talk I will describe current efforts to classify and analyze low dimension homogeneous spaces using the computer algebra system Maple. Applications to general relativity and to the exact integration of both ordinary and partial differential equations will be given.
The basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation for natuaral Hamiltonian systems are illustrated. In particular the geometrical properties of Killing tensors and conformal Killing tensors related to the separation are examined.
A method due to E. Cartan is used to give a classification of the simply connected four dimensional non-reductive homogeneous pseudo-Riemannian manifolds.
We present some examples of pseudo-Riemannian manifolds which are curvature homogeneous but not locally homogeneous. All standard local invariants vanish for these manifolds. These manifolds are Osserman, Ivanov-Petrov, and Stanilov. Some are modeled on irreducible symmetric spaces. We also exhibit k curvature homogeneous manifolds for arbitrarily large values of k.
In this talk we will discuss negatively curved homogeneous spaces admitting a simply transitive group of isometries (or equivalently, left-invariant metrics on Lie groups). Negatively curved spaces have a remarkably rich and diverse structure and are interesting from both a mathematical and a physical perspective. As well as giving general criteria for having left-invariant metrics with negative Ricci curvature scalar, we also consider special cases, like Einstein spaces and Ricci nilsolitons. We point out the relevance these spaces play in some higher-dimensional theories of gravity.
We study the algebraic aspects of the differential invariants as constructed by the moving frame (Fels & Olver (1999)). Contrary to the basic assumption of classical differential algebra (Riquier (1910), Ritt (1951), Kolchin (1973)) the derivations naturally acting on differential invariants satisfy some non trivial commutation rules. We generalize classical differential algebra and differential elimination to this setting.
This is part of a project initiated by E. Mansfield and in collaboration with I. Kogan. The initial goal was to treat systems that are symmetric under a Lie group action in a more appropriate frame. Other applications are within the scope of the Maple software developed. For instance finding a minimal set of generating differential invariants for a given Lie group action.
It was shown by S. Lie that almost every variational problem, invariant with respect to a group of point transformations, and the corresponding Euler-Lagrange equations, can be written in terms of differential invariants and invariant differential forms, and thus reduced by the symmetry group. In this talk we define infinitesimal variational symmetries of the reduced variational problem and show that they lead to conservation laws for both, the reduced and the original system of Euler-Lagrange equations. Computational aspects of this approach will be also discussed.
The gradient of the areal radius of an arbitrary spherically symmetric field can be spacelike, timelike and even null. A careful distinction of these possibilities is important and, for example, forms an essential element of any complete proof of the Birkhoff theorem. The possibility of spacelike and timelike gradients were studied extensively in the Russian literature (and labeled "R" and "T" regions respectively) some forty years ago and yet there appears to be no readily available extension of these ideas to an arbitrary spacetime. It is the purpose of this work to provide such an extension and to explore this extension away from spherical symmetry. This extension involves differential invariants of order three.
In this talk we will describe how to write geometric evolutions (general invariant evolutions of curves) as evolutions on the Lie algebra associated to general flat semisimple homogeneous spaces. We will also talk about differential invariants of curves on these spaces, moving coframes and about how to find algebraically the evolution induced on the differential invariants by these general geometric evolutions.
The Blaschke conjecture aims to classify the Riemannian manifolds whose cut loci are metric spheres-in other words, spaces in which light rays shooting out of a given point must all focus at the same time. There is very little known about the local geometry of these manifolds (e.g. sign of curvature is unknown), but globally we know a few things (e.g. all geodesics are periodic). Using Cartan's method of the moving frame, we get a simple proof that a Blaschke manifold with the same Betti numbers as a complex projective space must be diffeomorphic to a complex projective space.
In this talk I shall discuss Lorentzian spacetimes where all zeroth and first order curvature invariants vanish and show how this class differs from the VSI spacetimes. We show that for VSI spacetimes all components of the Riemann tensor and its derivatives up to some fixed order can be made arbitrarily small. This is illustrated by way of some examples.
I will report on my ongoing joint work with Peter Olver on developing systematic and constructive algorithms for the identification and analysis of various invariants for infinite dimensional pseudogroups. In this talk I will focus on techniques from commutative algebra combined with moving frames to discuss Tresse-Kumpera type existence results for differential invariants of a pseudogroup action.
We determine all VSI spacetimes (i.e. Lorentzian manifolds with vanishing curvature invariants) in arbitrary dimension higher then four. For this reason we need to generalize some methods developed in the context of General Relativity (e.g. the Petrov classification, the Newman-Penrose formalism) to higher dimensions. It turns out that the resulting VSI conditions are similar to the four-dimensional case.
This work was done in cooperation with A. Coley and R. Milson (Dalhousie) and A. Pravdova (Prague).
A curvature invariant of order n is a scalar obtained by contraction from a polynomial in the Riemann tensor and its covariant derivatives up to the order n. We determine all four dimensional Lorentzian manifolds for which all curvature invariants of all orders vanish (VSI spacetimes). There are 16 non-flat classes of such spacetimes, one of them being the well known pp-wave. All corresponding metrics may be given explicitly.
This work was done in cooperation with A. Coley and R. Milson (Dalhousie) and V. Pravda (Prague).
Multiple factors have contributed to the recent resurgence of interest in the classical invariant theory (CIT) of homogeneous polynomials among pure and applied mathematicians alike.
The factor that brought my collaborators and me to the area is that the underlying ideas of CIT can be naturally incorporated into the study of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The resulting theory shares many of the same essential features with CIT.
I target to review some of the results obtained so far. The project was initiated in a joint work with Ray McLenaghan and Dennis The in 2001.
We present an analysis of Killing tensors in the Euclidean plane using the moving frame method and derive invariants under the action of the isometry group. As an application, we demonstrate how these invariants are useful in the classification of the separable coordinate systems for the Hamilton-Jacobi equation.