Let (M,g) be an n-dimensional smooth Riemannian manifold without boundary, consider the Schouen tensor, a symmetric tensor defined as

A _g=(1/(n-2))(Ric-R/(2(n-1))g,

where Ric and R denote respectively the Ricci tensor and the scalar curvature associated with g.

Let f be some symmetric functions of $(λ ₁ ... λ_n)$, we study equation

f(μ(A _g))=1,

where μ(A _g) denotes the eigenvalues of A_g with respect to g.

We present some recent joint work with Aobing Li on such equations, which includes results on Harnack type inequality, Liouville type theorem, existence and compactness of solutions. One particular case is to take

f(λ)=σ _k(λ) :=Σ λ_i1 ... λ_ik. where the sum is over 0 < i1 < ... < ik< n+1.