  We study closed, simply connected manifolds with positive
$2^\mathrm{nd}$-intermediate Ricci curvature and large symmetry rank. In odd
dimensions, we show that they are spheres. In even dimensions other than $6$,
we show that they must have positive Euler characteristic. Under stronger
assumptions on the symmetry rank, we show that such even dimensional manifolds
must have trivial odd degree integral cohomology, and if the second Betti
number is no more than $1$, they are either spheres or complex projective
spaces. In the process, we establish new tools for studying isometric actions
on closed manifolds with positive $k^\mathrm{th}$-intermediate Ricci for values
of $k \geq 2$. These tools include generalizations of the isotropy rank lemma,
symmetry rank bound, and connectedness principle from the setting of positive
sectional curvature.
