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We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of τ-measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties of operators dominated in the sense of complete positivity by compact or by Dunford-Pettis operators. [ABSTRACT FROM AUTHOR]

Let X be a smooth curve on a ruled surface π : S → C. In this paper, we deal with the questions on the gonality and the Clifford index of X and on the composedness of line bundles on X with the covering morphism π∣X. The main theorem shows that if a smooth curve X ∼ aCo + bf satisfies some conditions on the degree of b, then a line bundle L on X with Cliff(L) ⩽ ag(C) - 1 is composed with π∣X. This implies that a part of the gonality sequence of X is computed by the gonality sequence of C as follows: dr(X) = adr(C) for r ⩽ L, where L is the length of the gonality sequence of C. [ABSTRACT FROM AUTHOR]

It is the aim of this article to give a reasonably detailed account of a specific bundle of geometric investigations and results pertaining to arithmetic groups, the geometry of the corresponding locally symmetric space X/Γ attached to a given arithmetic subgroup Γ ⊂ G of a reductive algebraic group G and its cohomology groups H*(X/Γ,ℂ). We focus on constructing totally geodesic cycles in X/Γ which originate with reductive subgroups H ⊂ G. In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group Γ is strongly related to the automorphic spectrum of Γ, this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms. [ABSTRACT FROM AUTHOR]

The article presents several reprints from mathematical reviews. One discusses on integral geometry in Klein spaces which is considered as a transitive Lie group G of automorphisms. Another one explores the geometry of G-structures specifically on its basic general problems of existence. Another one also explores on characteristic forms and geometric variants.

We construct $mathcal {F}$-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if $M=E# X$, where $E$ is a fiber bundle with structure group $G$ and a fiber admitting a $G$-invariant metric of non-negative sectional curvature and $X$ admits an $mathcal {F}$-structure with one trivial covering, then one can construct on $M$ a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. $operatorname {Vol}_K(M)=0$). As a corollary we prove that all the elements in the Spin cobordism ring can be represented by manifolds $M$ with $operatorname {Vol}_K (M)=0$. [ABSTRACT FROM AUTHOR]

In a compact orbifold, for a small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball. [ABSTRACT FROM AUTHOR]