In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.
We prove an analogue of the result of Hsiang and Kleiner for 4-dimensional compact orbifolds with positive curvature and an isometric S 1 action. Additionally, we prove that when π 1 (|O 4 |) = 0, then π orb 1 (O) provides a bound on the failure of Z-valued Poincaré Duality of |O|, and if π orb 1 (O) = 0, then Z-valued Poincré Duality holds for |O|.
Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs and graphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We characterize when twosided group digraphs and graphs are weakly and strongly connected and count connected components, using both an explicit elementary perspective and group actions. Our results and examples address four open problems posed by Iradmusa and Praeger that concern connectedness and valency. We pose five new open problems.
a r t i c l e i n f o a b s t r a c t One of the main methods of constructing new spaces with positive or almost positive curvature is the study of biquotients first studied in detail by Eschenburg. In this paper we classify orbifold biquotients of the Lie Group SU (3), and construct a new example of a 5-dimensional orbifold with almost positive curvature. Furthermore, we extend the work of Florit and Ziller on the geometric properties of the orbifolds SU (3)//T 2 .
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