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The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of N  ×  N matrices. The distributions are determined analytically: exactly for N  = 2 and approximately for N  ≥ 3, and compared with simulations. The distributions decay asymptotically as 1/ C 5/2 ; this is a consequence of the codimension of energy-level degeneracies in the ensemble.

In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qI−qη⊗ζ, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ-structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the J˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J˜ on FM. Finally, we construct an example of it to finish.

The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy Hγ,O is proportional to Rσ. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M, Hγ,O is an element of a Lie group G; the curvature Rσ∈g is an element of the Lie algebra of G. However, it turns out that the curvature form Rσ obtained from the small loop approximation is ambiguous, as the information of γ and Hγ,O are insufficient for determining a specific plane σ responsible for Rσ. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and f:Σ→M be an injective immersion. We prove that if the Lp functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.

A symmetric top is considered, which is a particular case of a mechanical top that is usually described by the canonical Poisson structure on T * SE (3). This structure is invariant under the right action of the rotation group SO(3), but the Hamiltonian of the symmetric top is invariant only under the right action of the subgroup S 1 , which corresponds to the rotation of the symmetric top around its axis of symmetry. This Poisson structure is obtained as the reduction T * SE (3) / S 1 . A Hamiltonian and motion equations are proposed that describe a wide class of interaction models of the symmetric top with an axially symmetric external field.

Two different charged dilaton black holes in 4-dimension, within teleparallel equivalent of general relativity (TEGR), are derived. These solutions are related through local Lorentz transformation. The total energy of these black holes, using three different methods, the Hamiltonian method, the translational momentum 2-form and the Euclidean continuation method given by Gibbons and Hawking, is calculated. It is shown that the three methods give the same results. The value of energy is shown to depend on the mass M and charge q . The verification of the first law of thermodynamics is proved. Finally, it is shown that if the charge q is vanishing then, the total energy reduced to that of Schwarzschild’s black hole.

Curvature tensors of Kaehler type (or type K) are defined on a hermitian vector space and it has been proved that the real vector space LK(V){\\mathcal {L}_K}(V) of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of V+{V^ + }, where VC=V+⊕V−{V^{\\text {C}}} = {V^ + } \\oplus {V^ - } (Theorem 3.6). Then it is shown that LK(V){\\mathcal {L}_K}(V) admits a natural orthogonal decomposition (Theorem 5.1) and hence every L∈LK(V)L \\in {\\mathcal {L}_K}(V) is expressed as L=L1+LW+L2L = {L_1} + {L_W} + {L_2}. These components are explicitly determined and then it is observed that LW{L_W} is a certain formal tensor introduced by Bochner. We call LW{L_W} the Bochner-Weyl part of L and the space of all these LW{L_W} is called the Weyl subspace of LK(V){\\mathcal {L}_K}(V).