Action Of Lie Group Research Articles

A vanishing theorem for the Mathai-Zhang index

We establish a vanishing theorem for spin manifolds admitting proper cocompact Lie group actions. To be more precise, when the connected component of the identity element in the action group is non-unimodular, we prove that the Mathai-Zhang index [Adv. Math. 225 (2010), pp. 1224–1247] vanishes.

Holomorphic Lie group actions on Danielewski surfaces

We prove that any Lie subgroup G (with finitely many connected components) of an infinite-dimensional topological group which is an amalgamated product of two closed subgroups can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).

Bismut–Cheeger Eta Form and Higher Spectral Flow

Abstract We prove the variation formula, the embedding formula, and the adiabatic limit formula for equivariant Bismut–Cheeger eta forms with a fiberwise compact Lie group action. We only need the condition that the kernels of corresponding fiberwise Dirac operators form vector bundles.

Expansive Lie Group Actions

In this work we introduce a concept of expansiveness for actions of connected Lie groups. We study some of its properties and investigate some implications of expansiveness. We study the centralizer of expansive actions and introduce CW-expansiveness for pseudo-group actions. As an application, we prove positiveness of geometric entropy for expansive foliations and expansive group actions.

Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the second case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus.

Classical Gauge Principle - From Field Theories to Classical Mechanics

In this paper we discuss how the gauge principle can be applied to classical-mechanics models with finite degrees of freedom. The local invariance of a model is understood as its invariance under the action of a matrix Lie group of transformations parametrized by arbitrary functions. It is formally presented how this property can be introduced in such systems, followed by modern applications. Furthermore, Lagrangians describing classical-mechanics systems with local invariance are separated in equivalence classes according to their local structures.

On cohomogeneity one Hermitian non-Kähler metrics

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.

Visible actions of compact Lie groups on complex spherical varieties

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.

Cohomogeneity one manifolds and homogeneous spaces of positive scalar curvature

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

Lifting spectral triples to noncommutative principal bundles

Given a free action of a compact Lie group G on a unital C⁎-algebra A and a spectral triple on the corresponding fixed point algebra AG, we present a systematic and in-depth construction of a spectral triple on A that is build upon the geometry of AG and G. We compare our construction with a selection of established examples.

Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

Quotients of singular foliations and Lie 2-group actions

Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M,\mathcal{F}_M) is the quotient of a foliated manifold (P,\mathcal{F}_P) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.

Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.

Fredholm conditions for operators invariant with respect to compact Lie group actions

Let G be a compact Lie group acting smoothly on a smooth, compact manifold M, let P∈ψ m (M;E 0 ,E 1 ) be a G–invariant, classical pseudodifferential operator acting between sections of two G-equivariant vector bundles E i →M, i=0,1, and let α be an irreducible representation of the group G. Then P induces a map π α (P):H s (M;E 0 ) α →H s-m (M;E 1 ) α between the α-isotypical components. We prove that the map π α (P) is Fredholm if, and only if, P is transversally α-elliptic, a condition defined in terms of the principal symbol of P and the action of G on the vector bundles E i .

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge–Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

A Splitting Result for Real Submanifolds of a Kähler Manifold

Let (Z,omega ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group U^mathbb {C}, where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U^mathbb {C} and let M be a G-invariant connected submanifold of Z. Let xin M. If G is a real form of U^mathbb {C}, we investigate conditions such that Gcdot x compact implies U^mathbb {C}cdot x is compact as well. The vice-versa is also investigated. We also characterize G-invariant real submanifolds such that the norm-square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of (Z,omega ) generalizing a result proved in Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).

A regular Lie group action yield smooth sections of the tangent bundle and relatedness of vector fields, diffeomorphisms

In this paper, we have concentrated on a group action on the tangent bundle of some smooth/differentiable manifolds which has been built from a regular Lie group action on such smooth/differentiable manifolds. Interestingly, elements of orbit space yield smooth sections of the tangent bundle having beautiful algebraic properties. Moreover, each of those smooth sections behaves nicely as a left-invariant vector field with respect to Lie group action by $G$. We have explained here a simple isomorphism between the set of such smooth sections and each tangent space of that smooth/differentiable manifold. Also we have discussed more about $F$-relatedness and have introduced vector field relatedness by notations $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. which are sets based on both vector field related diffeomorphisms and diffeomorphism related vector fields. We have presented consequences based on the algebraic structure on $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. sets and built some related group actions. We have placed some interrelationship between the both kinds of rel operations.

Bouquets revisited and equivariant elliptic cohomology

Let [Formula: see text] be an even-dimensional spin manifold with action of a compact Lie group [Formula: see text]. We review the construction of bouquets of analytic equivariant cohomology classes and of [Formula: see text]-integration. Given an elliptic curve [Formula: see text], we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on [Formula: see text]. Following Bott–Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten’s rigidity theorem. Our systematic use of the Chern–Weil construction of equivariant classes allows us to reduce proofs to algebraic identities.

Linear Dynamical Systems of Nilpotent Lie Groups

We study the topology of orbits of dynamical systems defined by finite-dimensional representations of nilpotent Lie groups. Thus, the following dichotomy is established: either the interior of the set of regular points is dense in the representation space, or the complement of the set of regular points is dense, and then the interior of that complement is either empty or dense in the representation space. The regular points are by definition the points whose orbits are locally compact in their relative topology. We thus generalize some results from the recent literature on linear actions of abelian Lie groups. As an application, we determine the generalized \(ax+b\)-groups whose \(C^*\)-algebras are antiliminary, that is, no closed 2-sided ideal is type I.

Equivariant eta forms and equivariant differential K-theory

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary-dimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick (2009) about the well-definedness of the push-forward map.