A symmetric 2-tensor canonically associated to Q-curvature and its applications

In this article, we define a symmetric 2-tensor canonically associated to Q-curvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricci tensor. This tensor can also be used to understand Chang-Gursky-Yang's theorem on 4-dimensional Q-singular metrics. Moreover, we show an Almost-Schur Lemma holds for Q-curvature, which gives an estimate of Q-curvature on closed manifolds.Comment: 14 pages, new remarks, references and acknowledgement added in the introductio

Constraining the Mass Scale of a Lorentz-Violating Hamiltonian with the Measurement of Astrophysical Neutrino-Flavor Composition

We study Lorentz violation effects on flavor transitions of high energy astrophysical neutrinos. It is shown that the appearance of Lorentz violating Hamiltonian can drastically change the flavor transition probabilities of astrophysical neutrinos. Predictions of Lorentz violation effects on flavor compositions of astrophysical neutrinos arriving on Earth are compared with IceCube flavor composition measurement which analyzes astrophysical neutrino events in the energy range between $25~{\rm TeV}$ and $2.8~{\rm PeV}$. Such a comparison indicates that the future IceCube-Gen2 will be able to place stringent constraints on Lorentz violating Hamiltonian in the neutrino sector. We work out the expected sensitivities by IceCube-Gen2 on dimension-$3$ CPT-odd and dimension-$4$ CPT-even operators in Lorentz violating Hamiltonian. The expected sensitivities can improve on the current constraints obtained from other types of experiments by more than two orders of magnitudes for certain range of the parameter space.Comment: Matches the published versio