Earthline Journal of Mathematical Sciences

On Quasi-Generalized ϕ-recurrent Para-Kenmotsu Manifolds

2026-07-23T16:12:18+00:00

The object of this paper is to study and reveal the various geometric properties of quasi-generalized $\phi-$recurrent (briefly, $\mathcal{QG}(\phi \mathcal{K}_{2n+1})$) para-Kenmotsu manifold. Among the results established here it is shown that a $\mathcal{QG}(\phi \mathcal{K}_{2n+1})$ para-Kenmotsu manifold is an Einstein manifold. Moreover, we prove that a Ricci soliton in a $\mathcal{QG}(\phi \mathcal{K}_{2n+1})$ para-Kenmotsu manifold is an expanding. Further, we study quasi-generalized $\mathcal{T}-\phi-$recurrent para-Kenmotsu manifold and obtain the results which reveal the nature of its associated 1-forms.

An Exponential-logarithmic Maximum Hardy-Hilbert-type Integral Inequality

2026-07-24T02:28:40+00:00

This paper introduces a new variant of the Hardy-Hilbert integral inequality, referred to as an exponential-logarithmic maximum Hardy-Hilbert-type integral inequality. Using this result, we establish an additional related inequality. For completeness and accessibility, the paper is self-contained, and all proofs are provided in detail.

Geometric Quantization of Coadjoint Superorbits and Schrödinger-Fock Representations of the Heisenberg Supergroup

2026-07-24T15:13:58+00:00

In the framework of Kirillov's orbit method for nilpotent supergroups, coadjoint superorbits of the Heisenberg supergroup naturally carry supersymplectic structures arising from the Kirillov-Kostant-Souriau form. Their geometric quantizationprovides a geometric realization of the irreducible unitary representations of the Heisenberg supergroup. In particular, after choosing an appropriate polarization, the resulting quantumstate space is identified with $L^2(\mathbb R^m)\otimes \Lambda^\bullet(\mathbb C^n),$ combining the bosonic Schrödinger representation with the fermionic Fock representation. Thus $\mathfrak h^{m|n}$ provides the simplest geometric model combining bosonic and fermionic quantization. The bosonic sector reproduces the canonical commutation relations, while the fermionic sector generates a Clifford algebra through anticommutation relations. This construction shows that Schrödinger-Fock representations arise naturally as the geometric quantization of coadjoint superorbits, thereby providing a deep connection between supergeometry, supersymplectic structures, and the representation theory of nilpotent supergroup. We begin by explicit calculations for the case $m=n=1$ and generalize to the arbitrary values of $m$ and $n$.