NONLINEAR MODELS IN 2 +<-DIMENSIONS

A generalization of the nonlinear $\ensuremath{\sigma}$ model is considered. The field takes values in a compact manifold $M$ and the coupling is determined by a Riemannian metric on $M$. The model is renormalizable in $2+\ensuremath{\epsilon}$ dimensions, the renormalization group acting on the infinite-dimensional space of Riemannian metrics. Topological properties of the $\ensuremath{\beta}$ function and solutions of the fixed-point equation ${R}_{\mathrm{ij}}\ensuremath{-}\ensuremath{\alpha}{g}_{\mathrm{ij}}={\ensuremath{\nabla}}_{i}{v}_{j}+{\ensuremath{\nabla}}_{j}{v}_{i}$, $\ensuremath{\alpha}=\ifmmode\pm\else\textpm\fi{}1 or 0$, are discussed.