In this thesis we extend a result of M.Walsh that showed that, under reasonable conditions, positive scalar curvature metrics which are Gromov-Lawson concordant are in fact isotopic. This thesis generalises this result by proving that Gromov–Lawson concordance implies isotopy in the space of Riemannian metrics on simply connected, smooth, closed manifolds with positive Ricci-(k, n) curvature for certain k at least 3 when n ≥ 5. To do this, we use a strengthening of the Gromov-Lawson surgery technique for extending a positive scalar curvature metric over the trace of a codimension ≥ 3 surgery to a positive scalar curvature metric which is a product near the boundary. We extend this to positive Ricci-(k, n) curvature metrics making use of theorems of Wolfson and Kordass. We also compute the Ricci-(k, n) curvature on a variety of standard metrics on the sphere and disc including so-called mixed torpedo metrics. In addition we give the conditions under which these standard metrics are isotopic in the space of positive Ricci-(k, n) curvature metrics. Moreover we extend a theorem of Carr to show that the space of positive Ricci- (k, 4n−1) curvature metrics on a (4n−1)-dimensional, smooth, closed, spin manifold, n ≥ 2, has infinitely many path components.