Search Results (24)

The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as an invariant Casimir function on symplectic leaves—specifically, the coadjoint orbits of the Lie group acting on the system, where these orbits are interpreted as level sets of entropy. We present a symplectic foliation interpretation of thermodynamics, based on Jean-Marie Souriau’s Lie group thermodynamics. This model offers a Lie algebra cohomological characterization of entropy, viewed as an invariant Casimir function in the coadjoint representation. The dual space of the Lie algebra is foliated into coadjoint orbits, which are identified with the level sets of entropy. Within the framework of thermodynamics, dynamics on symplectic leaves—described by the Poisson bracket—are associated with non-dissipative phenomena. Conversely, on the transversal Riemannian foliation (defined by the level sets of energy), the dynamics, characterized by the metric flow bracket, induce entropy production as transitions occur from one symplectic leaf to another. Full article

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes [...] Full article

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional $N=1,2,3$- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable $N=2,3$-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra $\mathcal{K}(1|3)$ and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the $N=1,2,3$-supercircle. Full article

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described. Full article

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows. Full article

Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey–Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a Bäcklund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions. Full article

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, $\forall x,y\in g$,$[x,y]=-[y,x]$, that is, the operation $[,]$ has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5$, or $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5{G}_6$, or $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5{G}_3$, where $G_i$ is a commutative subgroup of $Aut\left(g\right)$. We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups. Full article

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle $TG$ of $G.$ In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle $TG$, and we express in different cases the contravariant Levi-Civita connection and curvature of $TG$ in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms ${\Omega }^{*}(G)$ on G is a differential graded Poisson algebra if, and only if, ${\Omega }^{*}(TG)$ is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group $TG$ is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given. Full article

This paper defines a large class of differentiable manifolds that house two distinct optimal problems called affine-quadratic and rolling problem. We show remarkable connections between these two problems manifested by the associated Hamiltonians obtained by the Maximum Principle of optimal control. We also show that each of these Hamiltonians is completely intergrable, in the sense of Liouville. Finally we demonstrate the significance of these results for the theory of mechanical systems. Full article

A statistical transformation model consists of a smooth data manifold, on which a Lie group smoothly acts, together with a family of probability density functions on the data manifold parametrized by elements in the Lie group. For such a statistical transformation model, the Fisher–Rao semi-definite metric and the Amari–Chentsov cubic tensor are defined in the Lie group. If the family of probability density functions is invariant with respect to the Lie group action, the Fisher–Rao semi-definite metric and the Amari–Chentsov tensor are left-invariant, and hence we have a left-invariant structure of a statistical manifold. In the present work, the general framework of statistical transformation models is explained. Then, the left-invariant geodesic flow associated with the Fisher–Rao metric is considered for two specific families of probability density functions on the Lie group. The corresponding Euler–Poincaré and the Lie–Poisson equations are explicitly found in view of geometric mechanics. Related dynamical systems over Lie groups are also mentioned. A generalization in relation to the invariance of the family of probability density functions is further studied. Full article

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation. Full article

Ligand modification by substituting chemical groups within the binding pocket is a popular strategy for kinase drug development. In this study, a series of pteridin-7(8H)-one derivatives targeting wild-type FMS-like tyrosine kinase-3 (FLT3) and its D835Y mutant (FL3_D835Y) were studied using a combination of molecular modeling techniques, such as docking, molecular dynamics (MD), binding energy calculation, and three-dimensional quantitative structure-activity relationship (3D-QSAR) studies. We determined the protein–ligand binding affinity by employing molecular mechanics Poisson–Boltzmann/generalized Born surface area (MM-PB/GBSA), fast pulling ligand (FPL) simulation, linear interaction energy (LIE), umbrella sampling (US), and free energy perturbation (FEP) scoring functions. The structure–activity relationship (SAR) study was conducted using comparative molecular field analysis (CoMFA) and comparative molecular similarity indices analysis (CoMSIA), and the results were emphasized as a SAR scheme. In both the CoMFA and CoMSIA models, satisfactory correlation statistics were obtained between the observed and predicted inhibitory activity. The MD and SAR models were co-utilized to design several new compounds, and their inhibitory activities were anticipated using the CoMSIA model. The designed compounds with higher predicted pIC₅₀ values than the most active compound were carried out for binding free energy evaluation to wild-type and mutant receptors using MM-PB/GBSA, LIE, and FEP methods. Full article

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds. Full article

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter $k_{\mathbb{R}}$-spaces and locally convex spaces E such that $E\times E$ is a $k_{\mathbb{R}}$-space. Full article

Let $(X,G,{\omega }_1,{\omega }_2,\left\{{\eta }^t\right\})$ be a manifold with a bi-Poisson structure $\left\{{\eta }^t\right\}$ generated by a pair of G-invariant symplectic structures ${\omega }_1$ and ${\omega }_2$, where a Lie group G acts properly on X. We prove that there exists two canonically defined manifolds $(R_{L^i},G^i,{\omega }_1^i,{\omega }_2^i,\left\{{\eta }_i^t\right\})$, $i=1,2$ such that (1) $R_{L^i}$ is a submanifold of an open dense subset $X_{\left(H\right)}\subset X$; (2) symplectic structures ${\omega }_1^i$ and ${\omega }_2^i$, generating a bi-Poisson structure $\left\{{\eta }_i^t\right\}$, are $G^i$- invariant and coincide with restrictions ${\omega }_1{|}_{R_{L^i}}$ and ${\omega }_2{|}_{R_{L^i}}$; (3) the canonically defined group $G^i$ acts properly and locally freely on $R_{L^i}$; (4) orbit spaces $X_{\left(H\right)}/G$ and $R_{L^i}/G^i$ are canonically diffeomorphic smooth manifolds; (5) spaces of G-invariant functions on $X_{\left(H\right)}$ and $G^i$-invariant functions on $R_{L^i}$ are isomorphic as Poisson algebras with the bi-Poisson structures $\left\{{\eta }^t\right\}$ and $\left\{{\eta }_i^t\right\}$ respectively. The second Poisson algebra of functions can be treated as the reduction of the first one with respect to a locally free action of a symmetry group. Full article