Related papers: The 1856 Lemma of Cayley Revisited

The invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The…

We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss…

The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any…

The systematic derivation of constants of the motion, based on Killing tensors and the gauge covariant approach, is outlined. Quantum dots are shown to support second-, third- and fourth-rank Killing tensors.

We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum…

A classical theorem of Riemannian geometry, due in its original form to Cartan, states that the Taylor expansion of the metric in geodesic normal coordinates is a universal formal power series involving only the symmetrizations of the…

The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately.…

We give a proof of the KAM theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is…

We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need…

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…

We propose construction of a unique and definite metric ($\eta_+$), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, and CPT invariant and PT(CPT)-norm is indefinite (definite).…

We discuss Lie-Tresse theorem for the pseudogroup of diffeomorphisms acting on the space of (pseudo-)Riemannian metrics, and relate this to existence of Killing vector fields. Then we discuss the impact of symmetry in the general case.

We study the existence of a non-spacelike isometry, \zeta, in higher dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of…

The FRT quantum group and space theory is reformulated from the standard mathematical basis to an arbitrary one. The $N$-dimensional quantum vector Cayley-Klein spaces are described in Cartesian basis and the quantum analogs of…

The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply…

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of…

We present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for number fields and that "with moving targets" of Vojta) as an application of Geometric Invariant Theory (GIT). Roth's Theorem is…

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the…

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…

We study the local equivalence problem for Riemannian submersions under fiber-preserving isometries using differential invariants. After briefly recalling the vertical--horizontal splitting, the O'Neill tensors $A$ and $T$, and the mean…