  An important set of theorems in geometric analysis consists of constant rank
theorems for a wide variety of curvature problems. In this paper, for geometric
curvature problems in compact and non-compact settings, we provide new proofs
which are both elementary and short. Moreover, we employ our method to obtain
constant rank theorems for homogeneous and non-homogeneous curvature equations
in new geometric settings. One of the essential ingredients for our method is a
generalization of a differential inequality in a viscosity sense satisfied by
the smallest eigenvalue of a linear map (Brendle-Choi-Daskalopoulos, Acta Math.
219(2017): 1-16) to the one for the subtrace. The viscosity approach provides a
concise way to work around the well known technical hurdle that eigenvalues are
only Lipschitz in general. This paves the way for a simple induction argument.
