We find sufficient conditions for principal toric bundles over compact K¨ahler manifolds to admit Calabi-Yau connections with torsion, as well as conditions to admit strong K¨ahler connections with torsion. With the aid of a topological classification, we construct such geometry on (k−1)(S2×S4)#k(S3×S3) for all k ≥ 1.

One of our main results is the following: Let$X$be a compact connected subset of the Euclidean space$R$^{$n$}and$r(X, d$_{2}) the rendezvous number of$X$, where$d$_{2}denotes the Euclidean distance in$R$^{$n$}. (The rendezvous number$r(X, d$_{2}) is the unique positive real number with the property that for each positive integer$n$and for all (not necessarily distinct)$x$_{1},$x$_{2},...,$x$_{$n$}in$X$, there exists some$x$in$X$such that $$(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$$ .) Then there exists some regular Borel probability measure μ_{0}on$X$such that the value of ∫_{$X$}$d$_{2}($x, y$)$d$μ_{0}($y$) is independent of the choice$x$in$X$, if and only if$r$($X, d$_{2}) = sup_{μ}∫_{$X$}∫_{$X$}$d$_{2}($x, y$)$d$μ($x$)$d$μ($y$), where the supremum is taken over all regular Borel probability measures μ on$X$.

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.

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The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-K¨ahler manifolds with torsion