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Dec 2010

Volume 51, Issue 12, Articles (12xxxx)

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back to top Quantum Mechanics (General and Nonrelativistic)

Collocation method for fractional quantum mechanics

Paolo Amore, Francisco M. Fernández, Christoph P. Hofmann, and Ricardo A. Sáenz

J. Math. Phys. 51, 122101 (2010); http://dx.doi.org/10.1063/1.3511330 (16 pages)

Online Publication Date: 1 December 2010

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We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on little sinc functions, which discretizes the Schrödinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a Wentzel–Kramers–Brillouin analysis is performed.
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03.65.Ge Solutions of wave equations: bound states
02.60.-x Numerical approximation and analysis
02.30.Jr Partial differential equations

Quasiclassical asymptotics and coherent states for bounded discrete spectra

K. Górska, K. A. Penson, A. Horzela, G. H. E. Duchamp, P. Blasiak, and A. I. Solomon

J. Math. Phys. 51, 122102 (2010); http://dx.doi.org/10.1063/1.3503775 (12 pages) | Cited 1 time

Online Publication Date: 9 December 2010

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We consider discrete spectra of bound states for nonrelativistic motion in attractive potentials Vσ(x) = −|V0| |x|−σ, 0<σ ≤ 2. For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels eσ(n) of a bounded spectrum varying as eσ(n) ∼ −n−2σ/(2−σ). We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σ in the range 0 < σ ⩽ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l) with k and l positive integers satisfying k>l.
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03.65.Ge Solutions of wave equations: bound states
03.65.Sq Semiclassical theories and applications

The formal path integral and quantum mechanics

Theo Johnson-Freyd

J. Math. Phys. 51, 122103 (2010); http://dx.doi.org/10.1063/1.3503472 (31 pages)

Online Publication Date: 14 December 2010

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Given an arbitrary Lagrangian function on mathd and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by “Feynman diagrams,” although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a “Fubini theorem” expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by “cutting and pasting” and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic “formal path integral” for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.
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03.65.-w Quantum mechanics
03.65.Ge Solutions of wave equations: bound states
04.20.-q Classical general relativity

Wave functions and energy spectra for the hydrogenic atom in math3×M

Robert A. Van Gorder

J. Math. Phys. 51, 122104 (2010); http://dx.doi.org/10.1063/1.3520507 (12 pages)

Online Publication Date: 16 December 2010

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We consider the hydrogenic atom in a space of the form math3×M, where M may be a generalized manifold obeying certain properties. We separate the solution to the governing time-independent Schrödinger equation into a component over math3 and a component over M. Upon obtaining a solution to the relevant eigenvalue problems, we recover both the wave functions and energy spectrum for the hydrogenic atom over math3×M. We consider some specific examples of M, including the fairly simple D-dimensional torus TD and the more complicated Kähler conifold K in order to illustrate the method. In the examples considered, we see that the corrections to the standard energy spectrum for the hydrogen atom due to the addition of higher dimensions scale as a constant times 1/L2, where L denotes the size of the additional dimensions. Thus, under the assumption of small compact extra dimensions, even the first energy corrections to the standard spectrum will be quite large.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra

The U(1)-Kepler Problems

Guowu Meng

J. Math. Phys. 51, 122105 (2010); http://dx.doi.org/10.1063/1.3527268 (12 pages) | Cited 1 time

Online Publication Date: 22 December 2010

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Let n ⩾ 2 be a positive integer. To each irreducible representation σ of U(1), a U(1)-Kepler problem in dimension (2n − 1) is constructed and analyzed. This system is superintegrable and when n = 2 it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is math(n,n), and the Hilbert space of bound states H(σ) is the unitary highest weight representation of math(n,n) with the minimal positive Gelfand-Kirillov dimension. Furthermore, it is shown that the correspondence between σ* (the dual of σ) and H(σ) is the theta-correspondence for dual pair (U(1),U(n,n))⊆ Sp 4n(math).
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11.30.Na Nonlinear and dynamical symmetries (spectrum-generating symmetries)
11.10.St Bound and unstable states; Bethe-Salpeter equations
02.20.-a Group theory

Riemann surface approach to bound and resonant states for a nonlocal potential

Cornelia Grama, N. Grama, and I. Zamfirescu

J. Math. Phys. 51, 122106 (2010); http://dx.doi.org/10.1063/1.3527069 (12 pages)

Online Publication Date: 29 December 2010

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The Riemann surface approach to bound and resonant states is extended to the case of a separable nonlocal potential that is constant on a certain domain of the inner region and vanishes in the rest of the domain. The approach consists in the construction of the Riemann surface Rg of the S-matrix pole function k = k(g) over the g-plane, where g is the strength of the complex nonlocal potential. On the Riemann surface Rg the pole function k = k(g) is single-valued and analytic. The branch points of the pole function k = k(g) and their k-plane images are determined and analyzed as a function of the position of the region of nonlocality. The Riemann surface of the S-matrix pole function is constructed. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface Rg is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The nonlocal potential generates narrow resonant states that cannot be produced by a local potential.
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03.65.Nk Scattering theory
03.65.Ge Solutions of wave equations: bound states

On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs

Pavel Exner and Jiří Lipovský

J. Math. Phys. 51, 122107 (2010); http://dx.doi.org/10.1063/1.3526963 (19 pages)

Online Publication Date: 30 December 2010

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The subject of the paper is Schrödinger operators on tree graphs which are radial, having the branching number bn at all the vertices at the distance tn from the root. We consider a family of coupling conditions at the vertices characterized by (bn−1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {tn+1−tn} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger operator can be purely absolutely continuous.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ox Combinatorics; graph theory

Madelung representation of damped parametric quantum oscillator and exactly solvable Schrödinger–Burgers equations

Şirin A. Büyükaşık and Oktay K. Pashaev

J. Math. Phys. 51, 122108 (2010); http://dx.doi.org/10.1063/1.3524505 (27 pages)

Online Publication Date: 30 December 2010

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We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
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03.65.Ge Solutions of wave equations: bound states
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.30.Jr Partial differential equations
back to top Quantum Information and Computation

The χ2-divergence and mixing times of quantum Markov processes

K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf, and F. Verstraete

J. Math. Phys. 51, 122201 (2010); http://dx.doi.org/10.1063/1.3511335 (19 pages) | Cited 3 times

Online Publication Date: 14 December 2010

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We introduce quantum versions of the χ2-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in the literature for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore, the contractive behavior of the χ2-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.
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03.65.-w Quantum mechanics
02.60.-x Numerical approximation and analysis
02.50.Ga Markov processes

Universal coding for transmission of private information

Nilanjana Datta and Min-Hsiu Hsieh

J. Math. Phys. 51, 122202 (2010); http://dx.doi.org/10.1063/1.3521499 (20 pages)

Online Publication Date: 22 December 2010

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We consider the scenario in which Alice transmits private classical messages to Bob via a classical-quantum channel, part of whose output is intercepted by an eavesdropper Eve. We prove the existence of a universal coding scheme under which Alice's messages can be inferred correctly by Bob, and yet Eve learns nothing about them. The code is universal in the sense that it does not depend on specific knowledge of the channel. Prior knowledge of the probability distribution on the input alphabet of the channel, and bounds on the corresponding Holevo quantities of the output ensembles at Bob's and Eve's end suffice.
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03.67.Dd Quantum cryptography and communication security

From qubits to E7

Bianca Letizia Cerchiai and Bert van Geemen

J. Math. Phys. 51, 122203 (2010); http://dx.doi.org/10.1063/1.3519379 (25 pages)

Online Publication Date: 22 December 2010

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There is an intriguing relation between quantum information theory and super gravity, discovered by M. J. Duff and S. Ferrara. It relates entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant on the 56-dimensional representation of the Lie group E7. In this paper we recall the relatively straightforward manner in which three-qubits lead to E7, or at least to the Weyl group of E7. We also show how the Fano plane emerges in this context.
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03.67.Ac Quantum algorithms, protocols, and simulations
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
04.65.+e Supergravity
04.70.-s Physics of black holes
05.70.Ce Thermodynamic functions and equations of state

Chain of Hardy-type local reality constraints for n qubits

Sibasish Ghosh and Shasanka Mohan Roy

J. Math. Phys. 51, 122204 (2010); http://dx.doi.org/10.1063/1.3512994 (14 pages)

Online Publication Date: 29 December 2010

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Nonlocality without inequality is an elegant argument introduced by Hardy for two qubit systems, and later generalised to n qubits, to establish contradiction of quantum theory with local realism. Interestingly, for n = 2 this argument is actually a corollary of Bell-type inequalities, viz., the CH–Hardy inequality involving Bell correlations, but for n greater than two it involves n-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for n qubits and then associated with each such inequality, we provide a new Hardy-type local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'son-type theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
03.67.Lx Quantum computation architectures and implementations

The extension problem for partial Boolean structures in quantum mechanics

Costantino Budroni and Giovanni Morchio

J. Math. Phys. 51, 122205 (2010); http://dx.doi.org/10.1063/1.3523478 (22 pages)

Online Publication Date: 30 December 2010

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Alternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen–Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the “number of contexts,” classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole–Bell inequalities are indicated.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.30.-f Function theory, analysis
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Time-evolution of the external field problem in Quantum Electrodynamics

D.-A. Deckert, D. Dürr, F. Merkl, and M. Schottenloher

J. Math. Phys. 51, 122301 (2010); http://dx.doi.org/10.1063/1.3506403 (28 pages)

Online Publication Date: 7 December 2010

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See Also: Publisher's Note

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We construct the time-evolution for the second-quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g., Ruijsenaars) observed the Shale–Stinespring condition and showed that the one-particle time-evolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the time-evolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are well-defined and unique.
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12.20.Ds Specific calculations

Supersymmetry algebra cohomology. I. Definition and general structure

Friedemann Brandt

J. Math. Phys. 51, 122302 (2010); http://dx.doi.org/10.1063/1.3515844 (30 pages) | Cited 2 times

Online Publication Date: 7 December 2010

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This paper concerns standard supersymmetry algebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetry algebras, termed supersymmetry algebra cohomology, and corresponding “primitive elements” are defined by means of a BRST (Becchi-Rouet-Stora-Tyutin)-type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.
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11.30.Pb Supersymmetry
02.10.Ud Linear algebra
02.20.Sv Lie algebras of Lie groups

Quantized Nambu–Poisson manifolds and n-Lie algebras

Joshua DeBellis, Christian Sämann, and Richard J. Szabo

J. Math. Phys. 51, 122303 (2010); http://dx.doi.org/10.1063/1.3503773 (34 pages) | Cited 1 time

Online Publication Date: 9 December 2010

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We investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg n-Lie algebras in terms of foliations of mathn by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
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11.25.Yb M theory
02.30.Jr Partial differential equations
02.10.Ud Linear algebra
02.20.Sv Lie algebras of Lie groups
03.65.-w Quantum mechanics
02.40.-k Geometry, differential geometry, and topology

Klein–Gordon equation in hydrodynamical form

C. Y. Wong

J. Math. Phys. 51, 122304 (2010); http://dx.doi.org/10.1063/1.3526964 (15 pages)

Online Publication Date: 21 December 2010

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We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.
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11.10.Lm Nonlinear or nonlocal theories and models
03.65.Ge Solutions of wave equations: bound states
11.10.Cd Axiomatic approach

Hopf solitons in the Nicole model

Mike Gillard and Paul Sutcliffe

J. Math. Phys. 51, 122305 (2010); http://dx.doi.org/10.1063/1.3525805 (8 pages)

Online Publication Date: 22 December 2010

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The Nicole model is a conformal field theory in a three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme–Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.
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11.10.Lm Nonlinear or nonlocal theories and models
02.40.Pc General topology
11.25.Hf Conformal field theory, algebraic structures

Can we make a Bohmian electron reach the speed of light, at least for one instant?

Daniel V. Tausk and Roderich Tumulka

J. Math. Phys. 51, 122306 (2010); http://dx.doi.org/10.1063/1.3520529 (12 pages)

Online Publication Date: 28 December 2010

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In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light—a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current mathγμψ is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function ψ:math4→math4 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of math4 along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions ψ:math4→math4, Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.
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03.65.Ge Solutions of wave equations: bound states
03.65.Pm Relativistic wave equations
02.50.Cw Probability theory

Curvature function and coarse graining

Homero Díaz-Marín and José A. Zapata

J. Math. Phys. 51, 122307 (2010); http://dx.doi.org/10.1063/1.3521553 (20 pages) | Cited 1 time

Online Publication Date: 28 December 2010

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A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions have encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to “local details” ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function FS(A) defined below. It covers the holonomy function in the sense that expFS(A) = Hol (l = ∂S,A).
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11.15.-q Gauge field theories
02.20.Sv Lie algebras of Lie groups

Non-Abelian Berry phase, instantons, and N = (0,4) supersymmetry

João N. Laia

J. Math. Phys. 51, 122308 (2010); http://dx.doi.org/10.1063/1.3521497 (8 pages)

Online Publication Date: 30 December 2010

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In supersymmetric quantum mechanics, the non-Abelian Berry phase is known to obey certain differential equations. Here we study N = (0,4) systems and show that the non-Abelian Berry connection over R4n satisfies a generalization of the self-dual Yang–Mills equations. Upon dimensional reduction, these become the tt* equations. We further study the Berry connection in N = (4,4) theories and show that the curvature is covariantly constant.
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03.65.Ta Foundations of quantum mechanics; measurement theory
11.30.Pb Supersymmetry
12.40.Nn Regge theory, duality, absorptive/optical models
11.15.-q Gauge field theories
02.30.Hq Ordinary differential equations

Counting SO(9) × SU(2) representations in coordinate independent state space of SU(2) matrix theory

Yoji Michishita

J. Math. Phys. 51, 122309 (2010); http://dx.doi.org/10.1063/1.3527280 (8 pages)

Online Publication Date: 30 December 2010

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We consider decomposition of coordinate independent states into SO(9) × SU(2) representations in SU(2) matrix theory. To see what and how many representations appear in the decomposition, we compute the character, which is given by a trace over the coordinate independent states, and decompose it into the sum of products of SO(9) and SU(2) characters.
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11.30.Ly Other internal and higher symmetries
11.25.Yb M theory
back to top General Relativity and Gravitation

Past horizons in twisting algebraically special space–times

Włodzimierz Natorf

J. Math. Phys. 51, 122501 (2010); http://dx.doi.org/10.1063/1.3511331 (11 pages)

Online Publication Date: 14 December 2010

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Two equations describing past marginally trapped surfaces in twisting algebraically special space–times are obtained. One of them generalizes the equation discussed by Tod for twist-free (Robinson–Trautman) metrics. The second one is solvable under certain algebraic conditions, closely related to “m > 0” and “m2>a2” of the Kerr metric. Consequences of the existence of a null horizon are discussed. Kerr–Schild metrics admitting such horizons are shown to be of Petrov- type D.
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04.70.-s Physics of black holes
02.10.-v Logic, set theory, and algebra

Gravitational Chern–Simons and the adiabatic limit

Brendan McLellan

J. Math. Phys. 51, 122502 (2010); http://dx.doi.org/10.1063/1.3528667 (18 pages) | Cited 1 time

Online Publication Date: 28 December 2010

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We compute the gravitational Chern–Simons term explicitly for an adiabatic family of metrics using standard methods in general relativity. We use the fact that our base three-manifold is a quasiregular K-contact manifold heavily in this computation. Our key observation is that this geometric assumption corresponds exactly to a Kaluza–Klein Ansatz for the metric tensor on our three-manifold, which allows us to translate our problem into the language of general relativity. Similar computations have been performed by Guralnik et al. [Ann. Phys. 308, 222 (2008)], although not in the adiabatic context.
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04.50.Cd Kaluza-Klein theories
04.20.Gz Spacetime topology, causal structure, spinor structure
11.15.Yc Chern-Simons gauge theory
back to top Dynamical Systems

The warped Sasaki–Matsumoto metric and bundlelike condition

Y. Alipour-Fakhri and M. M. Rezaii

J. Math. Phys. 51, 122701 (2010); http://dx.doi.org/10.1063/1.3520636 (13 pages) | Cited 1 time

Online Publication Date: 21 December 2010

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Let mathn = (M,F) be a Finsler manifold and G be the Sasaki–Matsumoto metric on TM∘. Bejancu and Farran [“Finsler geometry and natural foliations on the tangent bundle,” Rep. Math. Phys. 58, 131 (2006)] proved that mathn = (M,F) is a Riemannian manifold if and only if the Sasaki–Matsumoto metric G on TM∘ is bundlelike for the vertical foliation. Let mathn1+n2 = (M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki–Matsumoto metric *G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then *G on TM∘ is bundlelike for the warped vertical foliation V*(TM∘) if and only if math1n1 = (M1,F1) and math2n2 = (M2,F2) are Riemannian manifolds.
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02.40.Hw Classical differential geometry
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