Curvature of a class of indefinite globally framed $f$-manifolds

Warped product contact CR-submanifolds of globally framed f-manifolds with Lorentz metric

In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if $M = N^{\perp} \times_f N^T$ is a warped CR-submanifold such that $N^{\perp}$ is $ϕ$?-anti-invariant and NT is $ϕ$?-invariant, then M is a CR-product. We show that the second fundamental form of a contact CR warped product of a indefinite S space form satisfies a geometric inequality, $\|h\|^2 \geq p{3\|\nabla lnf\|^2 - \triangle lnf + (c + 2)k + 1}$.

Indefinite Kasparov modules and pseudo-Riemannian manifolds

We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric), the harmonic oscillator, and the construction via the Kasparov product of an indefinite spectral triple from a family of spectr…

An Osserman-type condition on $g.f.f$-manifolds with Lorentz metric

A condition of Osserman type, called $ϕ$-null Osserman condition, is introduced and studied in the context of Lorentz globally framed $f$-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz $\mathcal{S}$-manifolds. We prove that a Lorentz $\mathcal{S}$-manifold with constant $ϕ$-sectional curvature is $ϕ$-null Osserman, extending a result stated for Lorentz Sasaki space forms. Then we state some characterizations for a particular class of $ϕ$-null Osserman $\cal{S}$-manifolds. Finally, some examples are examined.

Gupta-Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero mode…

芬斯勒光凸边界：在因果简单性与锥测地线空间 $\mathcal{N}$ 中的应用

Finslerian lightconvex boundaries: applications to causal simplicity and the space of cone geodesics $\mathcal{N}$

本文研究带边界的非定芬斯勒流形和锥结构，证明边界光凸性、内部因果简单性与锥测地线空间豪斯多夫性三者等价，并显式构造锥测地线空间的流形结构，推广了渐近 AdS 时空等应用。