Scalar Curvature, Injectivity Radius and Immersions with Small Second Fundamental Forms

Abstract

We prove in special cases the following. ${\bullet }_{Sc}$ Bounds on the {\it injectivity radii} of "topologically complicated" Riemannian $n$-manifolds $X$, where the scalar curvatures of $X$ are bounded from below, $Sc(X)\ge \sigma >0$. ${\bullet }_{curv}$ Lower bounds on {\it focal radii} of smooth immersions from $k$-manifolds, e.g. homeomorphic to the $k$-torus, to certain Riemannian manifolds of dimensions $n=k+m$, e.g. to the cylinders $S^{n-1}\times {\mathbb{R}}^1$. ${\bullet }_{mean}$ Topological lower bounds on the mean curvatures of domains in Riemannian manifolds. e.g. in the Euclidean $n$-space ${\mathbb{R}}^n$. At the present moment, our results are limited by the {\it spin condition} and the {\it $n\le 8$ restriction.}

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