In this dissertation, we study manifolds that have positive kth-intermediate Ricci curvature, which we abbreviate as Ric_k>0. This condition interpolates between having positive sectional curvature and having positive Ricci curvature. Specifically, we study this curvature condition in the presence of isometric group actions.

First, we show that if M is a positively curved homogeneous space, then M x M admits a metric with Ric_2>0. We also construct metrics with Ric_2>0 on several other products of homogeneous spaces. It follows from these examples that the Hopf Conjectures, Petersen-Wilhelm Conjecture, Berger Fixed Point Theorem, and Hsiang-Kleiner Theorem for positive sectional curvature do not hold in the setting of Ric_2 > 0.

Second, we establish the following: In a manifold with Ric_k>0, if a submanifold N has a tangential subspace on which the intrinsic kth-intermediate Ricci curvatures are non-positive, then the dimension of that subspace is bounded above by codim(N) + k. As a consequence, we obtain a local symmetry rank bound for manifolds with Ric_k>0.

Finally, we establish that if there are k+1 commuting Killing fields on a compact manifold with Ric_k>0, then there exists a point at which the Killing fields are linearly dependent. Using this, we establish a cohomogeneity obstruction and a symmetry rank bound for compact manifolds with Ric_k>0.