* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

The Geometry of Walker Manifolds

In complex hyperbolic geometry we consider an open set biholomorphic to an open ball in Cn, and we equip it with a particular metric that makes it have constant negative holomorphic curvature. This is analogous to but different from the real hyperbolic space. In the complex case, the sectional curvature is constant on complex lines, but it changes when we consider real 2-planes which are not complex lines.

Complex Hyperbolic Geometry

Suppose that we havethe unit Euclidean ball in Rn and construct new bodies using three op- erations — linear transformations, closure in the radial metric, and multiplicative summation defined by kxkK+0L = p kxkKkxkL. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p 6 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in everydimension. We provideFourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

The geometry of $L_0$

In this paper we present an axiomatic analysis of several ranking methods for general tournaments. We find that the ranking method obtained by applying maximum likelihood to the (Zermelo-)Bradley-Terry model, the most common method in statistics and psychology, is one of the ranking methods that perform best with respect to the set of properties under consideration. A less known ranking method, generalised row sum, performs well too. We also study, among others, the fair bets ranking method, widely studied in social choice, and the least squares method.

Paired comparisons analysis: an axiomatic approach to ranking methods

*-Ricci solitons of real hypersurfaces in non-flat complex space forms

We present the classification of all real hypersurfaces in complex hyperbolic space $\mathbb{C}H^{n}$, $n \geq 3$, with three distinct constant principal curvatures

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Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces

We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones. © 2007 American Mathematical Society.

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Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M . A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F , and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1 Isometric actions on the real hyperbolic plane The special linear group G = SL2(R) = {( a b c d )∣∣∣∣ a, b, c, d ∈ R, ad− bc = 1} acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form ( a b c d ) · z = az + b cz + d . By equipping H with the Riemannian metric ds = dx + dy y2 = dzdz =(z)2 (z = x+ iy)

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Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

José Carlos Díaz-Ramos

Papers

Real Hypersurfaces with Constant Principal Curvatures in Complex Hyperbolic Spaces

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

Homogeneous hypersurfaces in complex hyperbolic spaces

Almost Kähler Walker 4-manifolds