arXiv research

Expands Euler-Poincare characteristic to supergeometry.

Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X_0(Y_1)X_1 ... X_{n-1}(Y_n)X_n of a space Z. Among the applications is a succinct formula for the Euler-Poincare characteristic of Z, e(Z) = e(X_0) - e(Y_1) + e(X_1) - ... + e(X_{…

We prove a formula that relates the Euler-Poincaré characteristic of a closed semi-algebraic set to its Lipschitz-Killing curvatures

Extends transversality to supergeometry, proving stability and genericity.

Analyzes a finite set of metrics and functions to determine manifold torsion.

We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi's diastasis and its domain of definition, the Euler-Poincaré characteristic, the number of Borel-Morse cells, the Kodaira embedding theorem.

Two types of nonvanishing results are presented for compact Kähler varieties.

A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…

Let $\R$ be a real closed field, ${\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and ${\mathcal P} \subset \R[X_1,...,X_k]$ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebr…

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…

The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus a…

In this paper we provide a variational derivation of the Euler-Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the…

Develops higher-order Euler-Poincaré field equations for principal G-bundles.

Let $π:P\to M^n$ be a principal G-bundle, and let ${\mathcal{L}}: J^1P \toΛ^n(M)$ be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on ${\mathcal C}(P)$, the bundle of connections on P.

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…

We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a partial order $M \ge N \Leftrightarrow Mor (M, N) \neq φ$. For which $N$ the degrees of maps $f: M \t…

We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1,n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a for…

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singul…

In this paper we extend and Poincare dualize the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler-Poincare characteristic, to arbitrary manifolds. We use the Poincare dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray-Singer torsion by providing a sli…

Geometric derivation of quantum dynamics from Lie group actions.

The paper derives the QGS equations using stochastic central extensions.

The paper explores the topology of polygonal meshes and their properties.

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equatio…

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence…

Paper compares Lagrangian reduction methods for rigid body systems.

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Ca…

The action principle by Low [Proc. R. Soc. Lond. A 248, 282--287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the syst…

In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.

Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.

Unified product Lie groups and their quotient spaces are analyzed for dynamics.

We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.

Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…

We present in modern language the contents of the famous note published by Henri Poincaré in 1901 "Sur une forme nouvelle des équations de la Mécanique", in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equa…

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$ and 0 is a regular value of…

Extended orbit model theory for shape analysis using graded group action framework.

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism include…

We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V point-wise fixed. …

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $Π_1$ and $Π_2$ in constant contact angles $γ_1$ and $γ_2$ and bound, together with those planes, a…

We present an avatar of the Euler obstruction to foliated structures on certain non-metric surfaces. This adumbrates (at least for the simplest 2D-configurations) that the standard mechanism---to the effect that the devil of algebra sometimes barricades the existence of angelic geometric structures (obstruction theory …

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincaré equations on the matched pair of Lie algebra…

In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting appli…

In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to simultaneously cover several cases of interest in discrete and continuous descri…

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions …

Counterexamples to a conjecture on ribbon graph genus changes were found and proven.

New integrators for mechanical systems on Lie groups simplify based on group properties.

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $μ$ be the Euler-Poincaré measure on $PSL_2(\mathbb R)^n$ and $χ=μ/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…

We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove s…