In this paper we study behaviors of Legendre trajectories for Sasakian magnetic fields on real hypersurfaces in a nonflat complex space form. On real hypersurfaces of type (A) in a complex projective space, if we vary them along integral curves of characteristic vector fields, we do not have points where they are stational to the direction of integral curves. On real hypersurfaces of type (A) in a complex hyperbolic space, if the square of the strength of a magnetic field is greater than the absolute value of holomorphic sectional curvature of the ambient space, we still do not have such stational points, but we have such points along trajectories of principal curvature directions if the strength does not satisfies the condition.
After introducing the notion of generalized pseudo-anti commuting Ricci tensor defined by Ric phi + phi Ric = f phi for real hypersurfaces in the complex hyperbolic quadric Qm & lowast;, we give a complete classification of Hopf generalized Ricci-Bourguignon soliton real hypersurfaces in the complex hyperbolic quadric Qm & lowast;. As an application we obtain a classification of the gradient generalized Ricci-Bourguignon solitons on Hopf real hypersurfaces in Qm & lowast;.
We study almost gradient pseudo-Ricci solitons on Hopf hypersurfaces of non-flat complex space forms. We prove that such solitons are trivial gradient pseudo-Ricci solitons and that the hypersurface is pseudo-Einstein.
The (multi-)poly-Bernoulli numbers are defined by the generating series using the (multi-)polylogarithm series. Arakawa-Kaneko multiple zeta functions and Kaneko-Tsumura multiple zeta functions are known as generalizations of zeta functions, whose values at non-positive integers interpolate multi-poly-Bernoulli numbers. In this article, as an extension of (multi-)poly-Bernoulli numbers, inspired by Nakasuji-Phuksuwan-Yamazaki [6], we introduce Schur type multi-poly-Bernoulli numbers. Further, we give the expression of Schur type Arakawa-Kaneko multiple zeta functions in terms of Schur type multi-poly-Bernoulli numbers. Moreover, under the restriction to the hook type, we introduce hook type Kaneko-Tsumura multiple zeta functions and obtain expressions in terms of hook type multi-poly-Bernoulli numbers.
Fix a prime number p and an abelian field k. Let Uko (resp. Cko) be the projective limit of the semi-local units at p (resp. of the cyclotomic units) of the cyclotomic Zp-extension ko/k. We prove that the mu- invariant of the Zp[[T]]-module Uko /Cko is zero for all p and k. We also show that, for p = 2 and any imaginary abelian field k, the mu-invariant of (Uko /Cko)+ is [k : Q]/2 where (Uko /Cko)+ denotes the maximal quotient of Uko/Cko on which the complex conjugation acts trivially.
