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1.
Let (M,g)(M,g) be a noncompact complete Bach-flat manifold with positive Yamabe constant. We prove that (M,g)(M,g) is flat if (M,g)(M,g) has zero scalar curvature and sufficiently small L2L2 bound of curvature tensor. When (M,g)(M,g) has nonconstant scalar curvature, we prove that (M,g)(M,g) is conformal to the flat space if (M,g)(M,g) has sufficiently small L2L2 bound of curvature tensor and L4/3L4/3 bound of scalar curvature.  相似文献   

2.
We consider a Schrödinger-type differential expression HV=∇∇+VHV=+V, where ∇ is a Hermitian connection on a Hermitian vector bundle EE over a complete Riemannian manifold (M,g)(M,g) with metric gg and positive smooth measure dμdμ, and VV is a locally integrable section of the bundle of endomorphisms of EE. We give a sufficient condition for mm-accretivity of a realization of HVHV in L2(E)L2(E).  相似文献   

3.
A curve αα immersed in the three-dimensional sphere S3S3 is said to be a Bertrand curve if there exists another curve ββ and a one-to-one correspondence between αα and ββ such that both curves have common principal normal geodesics at corresponding points. The curves αα and ββ are said to be a pair of Bertrand curves in S3S3. One of our main results is a sort of theorem for Bertrand curves in S3S3 which formally agrees with the classical one: “Bertrand curves in S3S3 correspond to curves for which there exist two constants λ≠0λ0 and μμ such that λκ+μτ=1λκ+μτ=1”, where κκ and ττ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S3S3 as the only twisted curves in S3S3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S3S3 and (1,3)-Bertrand curves in R4R4.  相似文献   

4.
We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Zp)C(Zp) gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a nn dimensional vector space which we call HnHn. The ZpZp gauge particles act on the vertex particles and thus HnHn can be thought of as a C(Zp)C(Zp) module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of nn and pp, though we believe this feature holds for all n>pn>p. We will see that non-Abelian anyons of the quantum double of C(S3)C(S3) are obtained as part of the vertex excitations of the model with n=6n=6 and p=3p=3. Ising anyons are obtained in the model with n=4n=4 and p=2p=2. The n=3n=3 and p=2p=2 case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than ZpZp. This makes them possible candidates for realizing quantum computation.  相似文献   

5.
Even though the one-dimensional (1D) Hubbard model is solvable by the Bethe ansatz, at half-filling its finite-temperature T>0T>0 transport properties remain poorly understood. In this paper we combine that solution with symmetry to show that within that prominent T=0T=0 1D insulator the charge stiffness D(T)D(T) vanishes for T>0T>0 and finite values of the on-site repulsion UU in the thermodynamic limit. This result is exact and clarifies a long-standing open problem. It rules out that at half-filling the model is an ideal conductor in the thermodynamic limit. Whether at finite TT and U>0U>0 it is an ideal insulator or a normal resistor remains an open question. That at half-filling the charge stiffness is finite at U=0U=0 and vanishes for U>0U>0 is found to result from a general transition from a conductor to an insulator or resistor occurring at U=Uc=0U=Uc=0 for all finite temperatures T>0T>0. (At T=0T=0 such a transition is the quantum metal to Mott-Hubbard-insulator transition.) The interplay of the ηη-spin SU(2)SU(2) symmetry with the hidden U(1)U(1) symmetry beyond SO(4)SO(4) is found to play a central role in the unusual finite-temperature charge transport properties of the 1D half-filled Hubbard model.  相似文献   

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A multi-parametric version of the nonadditive entropy SqSq is introduced. This new entropic form, denoted by Sa,b,rSa,b,r, possesses many interesting statistical properties, and it reduces to the entropy SqSq for b=0b=0, a=r:=1−qa=r:=1q (hence Boltzmann–Gibbs entropy SBGSBG for b=0b=0, a=r→0a=r0). The construction of the entropy Sa,b,rSa,b,r is based on a general group-theoretical approach recently proposed by one of us, Tempesta (2016). Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of Sa,b,rSa,b,r with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy Sa,b,rSa,b,r can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles NN of the system, or even stabilizes, by increasing NN, to a limiting value.  相似文献   

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We continue the study of U(1)U(1) vortices with cholesteric vacuum structure. A new class of solutions is found which represent global vortices of the internal spin field. These spin vortices are characterized by a non-vanishing angular dependence at spatial infinity, or winding. We show that despite the topological Z2Z2 behavior of SO(3)SO(3) windings, the topological charge of the spin vortices is of the ZZ type in the cholesteric. We find these solutions numerically and discuss the properties derived from their low energy effective field theory in 1+11+1 dimensions.  相似文献   

10.
A complex symplectic structure on a Lie algebra hh is an integrable complex structure JJ with a closed non-degenerate (2,0)(2,0)-form. It is determined by JJ and the real part ΩΩ of the (2,0)(2,0)-form. Suppose that hh is a semi-direct product g?Vg?V, and both gg and VV are Lagrangian with respect to ΩΩ and totally real with respect to JJ. This note shows that g?Vg?V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of ΩΩ and JJ are isomorphic.  相似文献   

11.
We consider a complete nonnegative biminimal   submanifold MM (that is, a complete biminimal submanifold with λ≥0λ0) in a Euclidean space ENEN. Assume that the immersion is proper  , that is, the preimage of every compact set in ENEN is also compact in MM. Then, we prove that MM is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0λ<0, we construct examples of biminimal submanifolds and curves.  相似文献   

12.
In this paper we show that for a compact minimal hypersurface MM of constant scalar curvature in the unit sphere S6S6 with the shape operator AA satisfying ‖A‖2>5A2>5, there exists an eigenvalue λ>10λ>10 of the Laplace operator of the hypersurface MM such that ‖A‖2=λ−5A2=λ5. This gives the next discrete value of ‖A‖2A2 greater than 0 and 5.  相似文献   

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Suppose that the sphere SnSn has initially a homogeneous distribution of mass and let GG be the Lie group of orientation preserving projective diffeomorphisms of SnSn. A projective motion of the sphere, that is, a smooth curve in GG, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of SnSn and, more generally, examples of subgroups HH of GG such that a force free motion initially tangent to HH remains in HH for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=SOn+1H=SOn+1). The main tool is a Riemannian metric on GG, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.  相似文献   

15.
Motivated by the sigma model limit of multicomponent Ginzburg–Landau theory, a version of the Faddeev–Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space MM is an oriented Riemannian manifold and the target space NN is a Kähler manifold, and its properties are compared with the usual, uncoupled Faddeev–Skyrme model. In the case N=S2N=S2, it is shown that supercurrent coupling destroys the familiar topological lower energy bound of Vakulenko and Kapitanski when M=R3M=R3, and the less familiar linear bound when MM is a compact 3-manifold. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. In particular, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius RR is unstable   for all RR. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the usual Faddeev–Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low energy regime of two-component superconductors.  相似文献   

16.
Let uu be a function of nn independent variables x1,…,xnx1,,xn, and let U=(uij)U=(uij) be the Hessian matrix of uu. The symplectic Monge–Ampère equation is defined as a linear relation among all possible minors of UU. Particular examples include the equation detU=1detU=1 governing improper affine spheres and the so-called heavenly equation, u13u24u23u14=1u13u24u23u14=1, describing self-dual Ricci-flat 44-manifolds. In this paper we classify integrable symplectic Monge–Ampère equations in four dimensions (for n=3n=3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(uij)=0F(uij)=0 in more than three dimensions is necessarily of the symplectic Monge–Ampère type.  相似文献   

17.
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n)SO(m,n) symmetry as an isometry on an m+nm+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n)G(m,n) preserving a symmetry of the same type at dimension m+n−1m+n1, e.g.   a G(m,n−1)G(m,n1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4)SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3)G(1,3) may be relevant to real physics.  相似文献   

18.
We discuss three Hamiltonians, each with a central-field part H0H0 and a PT-symmetric perturbation igzigz. When H0H0 is the isotropic Harmonic oscillator the spectrum is real for all gg because HH is isospectral to H0+g2/2H0+g2/2. When H0H0 is the Hydrogen atom then infinitely many eigenvalues are complex for all gg. If the potential in H0H0 is linear in the radial variable rr then the spectrum of HH exhibits real eigenvalues for 0<g<gc0<g<gc and a PT phase transition at gcgc.  相似文献   

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