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

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, $Sc(X)\geq \sigma>0$. $\bullet_{curv}$ 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 $n\leq 8$ restriction.}