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Quantum Riemannian geometry of phase space and nonassociativity

Beggs E. J., Majid S.

Demonstratio Mathematica

2017

Vol. 50, nr 1

83--93

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.

On three-dimensional locally phi-recurrent quasi-Sasakian manifolds

De U., Sarkar A.

Demonstratio Mathematica

2008

Vol. 41, nr 3

677-684

The object of the present paper is to study three-dimensional locally phi-recurrent quasi-Sasakian manifolds.

Semi-slant submanifolds of t-manifolds

Khan V., Khan M.

Demonstratio Mathematica

2006

Vol. 39, nr 4

907-918

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.

On the geometry of hypersurfaces in a quasi-Sasakian manifold

Călin C.

Demonstratio Mathematica

2003

Vol. 36, nr 2

451--462

Harmonicity on nearly trans-Sasaki manifolds

Gherghe C.

Demonstratio Mathematica

2000

Vol. 33, nr 1

151-157

In this paper we shall study the harmonicity and D-pluriharmonicity of a (phi, J)-holomorphic map from a nearly trans-Sasaki manifold into an almost Hermitian manifold.