Symmetric Rearrangement and Geometric Inequalities on Riemannian Manifolds

This paper starts by introducing results from geometric measure theory to prove symmetric decreasing rearrangement inequalities on ${\mathbb{R}}^n$, which give multiple proofs of the isoperimetric and Pólya-Szegő inequalities. Then we consider smooth oriented Riemannian manifolds of the form $M^n=(0,\mathrm{\infty })\times {\mathrm{\Sigma }}^{n-1}$, and test what results carry over from the ${\mathbb{R}}^n$ setting or what assumptions about $M^n$ need to be added. Of particular interest was proving the smooth co-area formula in the Riemannian manifolds setting and re-formulating particular geometric inequalities.