Related papers: A variational principle for Kaluza-Klein type theo…

We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\rm GL}_N$ Lie group. The deformations are generated by $R$-matrices satisfying the associative Yang-Baxter equation. Using the coefficients…

Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the…

This article is founded on two fundamental principles: the principle field equations introduced in Refs. \cite{S, S1, S2} and the Fock-Ivanenko covariant derivatives \cite{FI, F}. The former yields the equations of motion for free fields of…

We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the…

The space M_n of all isomorphism classes of n-dimensional Lie algebras over a field k has a natural non-Hausdorff topology, induced from the Segal topology by the action of GL(n). One way of studying this complicated space is by topological…

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on $C^\infty(M)\otimes M_2(\mathbb{C})$ in terms of classical Riemannian geometry on a smooth manifold $M$, a finite quantum geometry…

In this note, we use give some algebraic applications of a previous result by the author which compares the deformations parameterized by the Maurer-Cartan elements of a differential graded Lie algebra, and a differential graded Lie…

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed…

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We…

We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo-Riemannian manifolds. We derive the first variational formulae…

We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…

Let $M$ be a fixed compact oriented embedded submanifold of a manifold $\overline{M}$. Consider the volume $\mathcal{V} (\overline{g}) = \int_M \mathsf{vol}_{(M, g)}$ as a functional of the ambient metric $\overline{g}$ on $\overline{M}$,…

Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm…

Let X_0 be a compact connected Riemann surface of genus g with D_0\subset X_0 an ordered subset of cardinality n, and let E_G be a holomorphic principal G-bundle on X_0, where G is a complex reductive affine algebraic group, that admits a…

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations…

We use a new variational method --based on the theory of anti-selfdual Lagrangians developed in [2] and [3]-- to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in $\R^{2N}$.…

Quantum spaces with $\frak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential $^*$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with…

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of…

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data…