In this talk, I will discuss about smooth higher rank lattice actions on manifolds with positive entropy. Especially, we see how one can construct a “homogeneous structure” on the manifold from dynamics. Then, for instance, when lattices in SL(n,R) (with n>2) act on an n-dimensional manifold with positive entropy, we can see that the lattice is commensurable with SL(n,Z) and some information about manifolds. This is joint work with Aaron Brown.