Cohomological tautness for Riemannian foliations

In this paper, we present new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.     

Conformal foliations and constraint quantization

We show that the Classical Constraint Algebra of a Parametrized Relativistic Gauge System induces a natural structure of Conformal Foliation on a Transversal Gauge. Using the theory of conformal foliations, we provide a natural Factor Ordering for the quantum operators associated to the canonical quantization of such gauge system.     

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations. Mathematics Subject Classifications (2000) 58J40, 58J42, 58B34.     

Stability between foliations in general relativity

The aim of this paper is to study foliations that remain invariant under parallel transport along the integral curves of vector fields of another foliation. According to this idea, we define a new concept of stability between foliations. A particular case of stability (called regular stability) is studied, giving a useful characterization in terms of the Riemann curvature tensor. This characterization allows us to prove that there are no regularly self-stable foliations of dimension greater than 1 in the Schwarzschild and Robertson–Walker space–times. Finally, we study the existence of regularly self-stable foliations in other space–times, like pp-wave space–times.     

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck-Lichnerowicz type formula which allows us to apply the classical Bochner-Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.     

Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.     

Symbolic Dynamics for the Tél Map

In terms of contracting and expanding foliations the two-dimensional Tél map is decomposed into two coupled one-dimensional maps. Symbolic sequences are assigned to the two classes of foliations, and their ordering is discussed. The pruning front is constructed. A necessary and sufficient condition for admissible sequences is proposed.     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Eigenvalues of the transversal Dirac operator on Kähler foliations

In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators.     

Global properties of supermanifolds

We construct new examples of supermanifolds, and determine the vector bundle structure of the supermanifolds commonly used in physics. We show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts, and which is one of a nested sequence of foliations. We point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds. For example, a compact supermanifold with a single even dimension must have vanishing Euler characteristic. We also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations. By this mechanism, the global topology of a supermanifold can be used to impose physically interesting constraints on superfields. As an example, we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral.Enrico Fermi Fellow. Research supported by the NSF: PHY 83-01221, and the Department of Energy: DE AC 02-82-ER-40073     

Asymptotics of action variables near semi-toric singularities

The presence of focus–focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action–Angle coordinates on such systems. At focus–focus critical points, the Liouville–Arnold–Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behavior and the multi-valuedness of the Action integrals is given by a complex logarithm. This extends a previous result by San Vũ Ngọc to any dimension. We also calculate the monodromy matrix for these systems.     

Convergence to Equilibrium for Intermittent Symplectic Maps

We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.     

Determination of Partition Lines from Dynamical Foliations for the Hénon Map

The binary partition lines for the Hhnon map are numerically constructed from tangencies between the contracting and expanding foliations. The ordering of foliations acording to their symbolic sequences are examined.     

A Dynamical Zeta Function for Pseudo Riemannian Foliations

We investigate a generalization of geodesic random walks to pseudo Riemannian foliations. The main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity, quantum mechanics and dynamical systems. Partially supported by DFG, SFB 478.     

Space-time foliations in gravitation theory

A relationship is established between gravitational fields and space-time foliations on a manifold X⁴; the gravitational singularities are described as singularities of these foliations representing critical points of real functions on X⁴.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 20–23, September, 1982.     

Some rigidity theorems for locally symmetrical Finsler manifolds

In this paper we study some rigidity properties for locally symmetrical Finsler manifolds and obtain some results. We obtain the local equivalent characterization for a Finsler manifold to be locally symmetrical and prove that any locally symmetrical Finsler manifold with nonzero flag curvature must be Riemannian. We also generalize a rigidity result due to Akbar-Zadeh.     

Spacelike metric foliations

In this paper, we investigate spacelike metric foliations in lightlike complete spacetimes. When such a foliation satisfies the strong energy condition Ric^V (e) ≥ 0 for timelike vectors e, it must be totally geodesic, and the metric is of higher rank, in the sense that each point of the spacetime is contained inside a flat, totally geodesic, timelike rectangle. If in addition Ric^V(e) = 0, then the metric is (at least locally) a product metric, with the leaves of the foliation tangent to one of the factors.     

How different are the supermanifolds of Rogers and DeWitt?

A DeWitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold, whereas a Rogers supermanifold is not so restricted. Corresponding to the vector space fibers of the DeWitt supermanifold, a Rogers supermanifold has a foliation by submanifolds, or leaves, parametrized by soul coordinates only. We show that the universal covering space of any leaf always admits a flat metric. If the covering space is complete in this metric, it must in fact be a vector space. We combine this result with known theorems about foliations to give conditions under which a compact Rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace. We also show that a supermanifold defined by a polynomial equation in flat superspace is always of the DeWitt type. Finally, we exhibit new supermanifold structures forR ² and the 2-torus which show that the foliation of a Rogers supermanifold can be quite exotic.Enrico Fermi Fellow. Research supported by the NSF: PHY 83-01221, and the Department of Energy: DE AC02-82-ER-40073     

Closed/open string diagrammatics

We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology.