The principal aim of this thesis is to establish a generalization of a classical result in positive scalar curvature due to Carr. This result asserts that given any smooth subcomplex of a Riemannian manifold with codimension at least three, there is a tubular neighbourhood whose boundary has an induced metric with positive scalar curvature. The aim is to generalize this to a range of stronger curvature conditions, namely k-positive Ricci curvature for k at least 2, Sck. The proposed theorem claims that for a dimension d subcomplex, there is a tubular neighbourhood boundary with (d+1)-positive Ricci curvature. This implies the Carr result when d is at most n − 3, where n is the dimension of the manifold. (While Carr’s result is not in doubt, his argument is problematic for various reasons, and offering a clear re-proof is of independent interest.) We illustrate the above theorem in two different ways. Firstly, we study the boundaries of plumbed manifolds, leading to a k-positive Ricci curvature generalization of a result of Crowley-Wraith for positive Ricci curvature. Secondly, we generalize a positive scalar curvature result of Carr concerning the fundamental group. Carr’s result claims that any finitely presented group is the fundamental group of a closed n-manifold with positive scalar curvature, for n ≥ 4. We show that the same statement holds if positive scalar curvature is replaced by Sc3 > 0. This result is also of interest in relation to a conjecture of Wolfson. The final chapter contains a separate project. A theorem of H. H. Wang shows that for a compact Riemannian manifold with boundary having Ric > 0 globally and non-negative sectional curvature at the boundary, if the boundary convexity is sufficiently high, the manifold must be contractible. We develop an alternative approach which allows an explicit estimate of the required boundary convexity to guarantee contractibility.