Data sets often have certain nonlinear structures. Modeling/Approximating the nonlinear structures by the manifold model is attracting more and more attention in data science nowadays. A natural question is to find coordinate charts for data sets, i.e., manifolds. In this talk, I will discuss embedding of manifolds via eigen-vector fields of the connection Laplacian. For data sets, the eigen-vector fields can be computed by the graph connection Laplacian (GCL). I will also discuss the mathematical framework of image denoising via the GCL.