In the class of real hypersurfaces M²n−¹ isometrically immersed into a nonflat complex space form Mn(c) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ℂPn(c) or a complex hyperbolic space ℂHn(c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in Mn(c), we consider a certain real hypersurface of type (A2) in ℂPn(c) and give a geometric characterization of this Hopf manifold., Byung Hak Kim, In-Bae Kim, Sadahiro Maeda., and Obsahuje bibliografii
We study some geometric properties associated with the t-geometric means A ♯_{t} B:= A^{1/2}(A^{-1/2}BA^{-1/2})^{t} A^{1/2}of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted., Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam., and Obsahuje seznam literatury
A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists., Tamás Milkovszki, Zoltán Muzsnay., and Seznam literatury