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

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.

Reduction and approximation in gyrokinetics

The gyrokinetics formulation of plasmas in strong magnetic fields aims at the elimination of the angle associated with the Larmor rotation of charged particles around the magnetic field lines. In a perturbative treatment or as a time-averaging procedure, gyrokinetics is in general an approximation to the true dynamics. Here we discuss the conditions under which gyrokinetics is either an approximation or an exact operation in the framework of reduction of dynamical systems with symmetryComment: 15 pages late