Hamiltonian reduction of free particle motion on the group SL(2, ℝ)

We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle motion over the groupSL(2, ℝ). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional planes or to a cylinder S¹×ℝ. We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic to the union of two cotangent bundles T*(ℝ) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle T* (S¹), which is also endowed with a canonical symplectic structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 149–161, January, 1997.     

Stabilization of 4-manifolds

The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.     

The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M ⁴ × ?¹. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, ?, ?, ?) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M ⁴ × S ¹, where S ¹ is the circle. For such functions, electric charges are topologically quantized.     

Nonsingularity in the no-boundary Universe

In the no-boundary Universe of Hartle and Hawking, the path integral for the quantum state of the Universe must be summed only over nonsingular histories. If the quantum corrections to the Hamilton-lacobi equation in the interpretation of the wave packet is taken into account, then all classical trajectories should be nonsingular. The quantum behaviour of the classical singularity in theS ¹×S ^m model (m⩾2) is also clarified. It is argued that the Universe should evolve from the zero momentum state, instead from a zero volume state, to a 3-geometry state.     

A note on certain square functions on product spaces

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition.     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators

We examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space ℝ² × T² in which there exist attracting, knotted 2-tori called torus braids. We develop a topological classification scheme in which a torus braid is characterized by closed braids that exist in two Poincare sections, ℝ² \times S¹ × {·} and ℝ² × {·} \times S¹. Based on the classification scheme, we develop numerical invariants that describe the linkedness of attractors and provide information about the global dynamics. Numerical simulations show that changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and locally "doubles," shedding a torus of different equivalence class. We call this a topological torus bifurcation of the doubling variety (TTBD). We provide a topological analysis of the doubling produced by TTBDs and we examine the qualitative dynamical changes that result. We also examine the effect of TTBDs on the spectrum of Lyapunov exponents and the time series power spectrum.     

New examples of Lagrangian rigidity

Letn=4 or 8. We prove that any Lagrangian embedding ofS ^{n − 1} ×S ¹ into ℂ ⁿ has a trivial linking class. We deduce that every embedding ofS ³ ×S ⁴ into ℂ⁴ is isotopic to a Lagrangian embedding. This is false ifn = 8.     

Rubber rolling over a sphere

“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G ₂). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S ² and base S ², that can be reduced to an almost Hamiltonian system in T*S ² with a non-closed 2-form ω_NH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I _j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω_NH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I _j are known.      

Further Restrictions on the Topology of Stationary Black Holes in Five Dimensions

We place further restriction on the possible topology of stationary asymptotically flat vacuum black holes in five spacetime dimensions. We prove that the horizon manifold can be either a connected sum of Lens spaces and “handles” S ¹ × S ², or the quotient of S ³ by certain finite groups of isometries (with no “handles”). The resulting horizon topologies include Prism manifolds and quotients of the Poincare homology sphere. We also show that the topology of the domain of outer communication is a cartesian product of the time direction with a finite connected sum of \mathbb R⁴,S² ×S²{\mathbb R^4,S^2 \times S^2} ’s and CP ²’s, minus the black hole itself. We do not assume the existence of any Killing vector beside the asymptotically time like one required by definition for stationarity.     

Classification des variété approximativement kähleriennes homogénes

A Riemannian homogeneous manifold admitting a strict nearly-Kähler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S³ × S³, S⁶, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kähler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrión. For the last two it is obtained by canonical variation of the Kähler structure of the twistor space over a four-dimensional manifold. Finally, from Bär, a nearly-Kähler structure on the sphere S⁶ corresponds to a constant 3-form on the Riemannian cone R⁷.Mathematics Subject Classifications (2000): 53C15, 53C25, 53C30, 53C56.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

Kaluza-Klein pistons with noncommutative extra dimensions

We calculate the scalar Casimir energy and Casimir force for an ℝ³ × N Kaluza-Klein piston setup in which the extra-dimensional space N contains a noncommutative two-dimensional sphere S _fz. We study the cases with T ^d ×S _fz and S _fz as the extra-dimensional spaces, where T ^d is the d-dimensional commutative torus, and examine the validity of the results and the regularization obtained in the piston setup in each case. We examine the Casimir energy with one-loop corrections for one piston chamber due to the self-interacting scalar field in the noncommutative geometry. We compute with some approximations. We compare the obtained results with the results of analogous computations for the Minkowski space-time M ^D . In conclusion, we discuss the stabilization of the extra-dimensional space in the piston setup.     

Cyclic gradings of Lie algebras and lax pairs for <Emphasis Type="Italic">σ</Emphasis>-models

We study a class of σ-models with complex homogeneous target spaces and zero-curvature representations. We find a relation between these models and σ-models with certain m-symmetric target spaces. We also describe a model with the hypercomplex target space S ¹ × S ³ in detail.     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Tight Lagrangian surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

We determine all tight Lagrangian surfaces in S ² × S ². In particular, globally tight Lagrangian surfaces in S ² × S ² are nothing but real forms of this symmetric space.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spin refined Kauffman bracket skein module ofS 1×S 2 and lens spaces

A method for computing spin refined skein modules of 3-manifolds from Heegaard splittings is described, and the spin refined skein modules of the lens spacesL(p, r) (includingL(0,1)=S ¹×S ²) are computed explicitly.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998