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Geometrical characterizations are given for the tensor R⋅SRS, where SS is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)(M,g) and RR denotes the curvature operator   acting on SS as a derivation, and of the Ricci Tachibana tensor  g⋅SgS, where the natural metrical operator  gg also acts as a derivation on SS. As a combination, the Ricci curvatures   associated with directions on MM, of which the isotropy determines that MM is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on MM, and of which the isotropy determines that MM is Ricci pseudo-symmetric in the sense of Deszcz.  相似文献   

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For every diffeomorphism φ:M→Nφ:MN between 3-dimensional Riemannian manifolds MM and NN, there are locally two 2-dimensional distributions D±D± such that φφ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of D±D±. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator induced from φφ. We investigate the integrability condition of D+D+ and DD. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.  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   

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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   

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We study reduction of generalized complex structures. More precisely, we investigate the following question. Let JJ be a generalized complex structure on a manifold MM, which admits an action of a Lie group GG preserving JJ. Assume that M0M0 is a GG-invariant smooth submanifold and the GG-action on M0M0 is proper and free so that MG?M0/GMG?M0/G is a smooth manifold. Under what condition does JJ descend to a generalized complex structure on MGMG? We describe a sufficient condition for the reduction to hold, which includes the Marsden–Weinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds.  相似文献   

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In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle VkVk to a decreasing family of kk foliations FiFi on a manifold MM. We have shown that there exists a (1,1)(1,1) tensor JJ of VkVk such that Jk≠0Jk0, Jk+1=0Jk+1=0 and we defined by LJ(Vk)LJ(Vk) the Lie Algebra of vector fields XX on VkVk such that, for each vector field YY on VkVk, [X,JY]=J[X,Y][X,JY]=J[X,Y].  相似文献   

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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   

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The distance d(i,j)d(i,j) between any two vertices ii and jj in a graph is the number of edges in a shortest path between ii and jj. If there is no path connecting ii and jj, then d(i,j)=∞d(i,j)=. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph (Latora and Marchiori, 2001) [1]. The efficiency between two vertices ii and jj is defined to be i,j=ji,j=j. In this paper, we investigate the efficiency of star-like networks, and show that networks of this type have a high level of efficiency. We apply these ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair of stations.  相似文献   

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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   

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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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