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

We prove special cases of the following. ●_Sc Bounds on the injectivity radii of “topologically complicated” Riemannian n-manifolds X, where the scalar curvatures of X are bounded from below, Sc(X) ≥ σ > 0. ●_curv Lower bounds on focal radii of smooth immersions from k-manifolds, e.g. homeo-morphic to the k-torus, to certain Riemannian manifolds of dimensions n = k +m, e.g. to the cylinders Sⁿ⁻¹ ×R¹. ●_mean Topological lower bounds on the mean curvatures of domains in Riemannian manifolds. e.g. in the Euclidean n-space Rⁿ. At the present moment, our results are mainly limited by the spin condition and the n ≤ 8 restriction with additional difficulties in the case of foliations. The corresponding problems and questions are presented in shades of red.