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Content available remote Quantum Riemannian geometry of phase space and nonassociativity
EN
Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of CPn where the commutation relations have the canonical form [wi, wj] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.
2
Content available remote On three-dimensional locally phi-recurrent quasi-Sasakian manifolds
EN
The object of the present paper is to study three-dimensional locally phi-recurrent quasi-Sasakian manifolds.
3
Content available remote Holomorphons on spheres
EN
We consider Euler–Lagrange equations of families of nonnegative functionals defined on tensor fields of the type (1, 1), which are equal to zero only for complex structures tensor fields.
4
Content available remote Semi-slant submanifolds of t-manifolds
EN
J. L. Cabrerizo et al. [5] studied slant submanifolds of Sasakian and K- contact manifolds. Semi-slant submanifolds were introduced as a generalized version of CR-submanifold. Cabrerizo et al. [4] obtained interesting results for the semi-slant submanifold of Sasakian manifolds. The purpose of the present paper is to study slant and semi-slant submanifolds of a T-manifold.
5
Content available remote Holomorphons and the standard almost complex structure on S^6
EN
We consider Euler–Lagrange equations of families of nonnegative functionals defined on tensor fields of the type (1, 1), which are equal to zero only for complex structures tensor fields. As a solution of the equations we define the notion of holomorphon to distinguish a new class of tensor fields on Riemannian manifolds. Next, as our main result, we construct a holomorphon on the 6–dimensional sphere S^6.
EN
In this paper we extend our considerations on the spin and twist motions in a homogenous continuum. Refering to our former results, we define the degenerated mechanics as that in which the displacement motions vanish while rotation motions, as the independent elastic fields, do exist. This is exactly the opposite case to the classic ideal elastic continuum in which only displacement motions are taken into account, while any independent rotation motions are excluded because of a lack of the appriopriate constitutive laws supporting the existence of an elastic response due to the rotational deformations of bonds in a lattice network. We propose a system of potentials which would help us to understand the waves and geometrical features of degenerated mechanics and its Riemannian geometry.
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