Extending tensors on polar manifolds

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $\Sigma\subset M$ and generalized Weyl group $W$. We show that restriction to $\Sigma$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $\Sigma$. Moreover, we show that every smooth $W$-invariant Riemannian metric on $\Sigma$ can be extended to a smooth $G$-invariant Riemannian metric on $M$ with respect to which the $G$-action remains polar with the same section $\Sigma$.Comment: arXiv admin note: text overlap with arXiv:1205.476

Comprehension of object-oriented software cohesion: The empirical quagmire

Chidamber and Kemerer (1991) proposed an object-oriented (OO) metric suite which included the Lack of Cohesion Of Methods (LCOM) metric. Despite considerable effort both theoretically and empirically since then, the software engineering community is still no nearer finding a generally accepted definition or measure of OO cohesion. Yet, achieving highly cohesive software is a cornerstone of software comprehension and hence, maintainability. In this paper, we suggest a number of suppositions as to why a definition has eluded (and we feel will continue to elude) us. We support these suppositions with empirical evidence from three large C++ systems and a cohesion metric based on the parameters of the class methods; we also draw from other related work. Two major conclusions emerge from the study. Firstly, any sensible cohesion metric does at least provide insight into the features of the systems being analysed. Secondly however, and less reassuringly, the deeper the investigative search for a definitive measure of cohesion, the more problematic its understanding becomes; this casts serious doubt on the use of cohesion as a meaningful feature of object-orientation and its viability as a tool for software comprehension

Sectional curvature and Weitzenb\"ock formulae

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also give an application of the Bochner technique to closed $4$-manifolds with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.