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## A gap phenomenon on Riemannian manifolds with reverse volume pinching

Acta Mathematica Hungarica
Author:
P. Wang

## Abstract

Let M n be an n(≧ 3)-dimensional compact, simply connected Riemannian manifold without boundary and S n be the unit sphere of the Euclidean space R n+1. By two different means we derive an estimate of the diameter whenever the manifold considered satisfies that the sectional curvature K M ≦ 1, while Ric (M) ≧
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{{n + 2}}{4}$$ \end{document}
and the volume V (M) ≦
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{3}{2}$$ \end{document}
(1 + η)V (S n ) for some positive number η depending only on n. Consequently, a gap phenomenon of the manifold will be given according to the estimate of the diameter.
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## A differentiable sphere theorem on manifolds with reverse volume pinching

Acta Mathematica Hungarica
Author:
P. Wang

## Abstract

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.

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## Three-dimensional semi-symmetric homogeneous Lorentzian manifolds

Acta Mathematica Hungarica
Author:
G. Calvaruso

## Abstract

Because of the different possible forms (Segre types) of the Ricci operator, semi-symmetry assumption for the curvature of a Lorentzian manifold turns out to have very different consequences with respect to the Riemannian case. In fact, a semi-symmetric homogeneous Riemannian manifold is necessarily symmetric, while we find some three-dimensional homogeneous Lorentzian manifolds which are semi-symmetric but not symmetric. The complete classification of three-dimensional semi-symmetric homogeneous Lorentzian manifolds is obtained.

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## Ricci solitons on Lorentzian Walker three-manifolds

Acta Mathematica Hungarica
Authors:
Giovanni Calvaruso
and
Barbara De Leo

## Abstract

We investigate Ricci solitons on Lorentzian three-manifolds (M,g f ) admitting a parallel degenerate line field. For several classes of these manifolds, described in terms of the defining function f, the existence of non-trivial Ricci solitons is proved.

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