Chern-Moser-Weyl tensor and embeddings into hyperquadrics

A central problem in Mathematics is the classification problem. Given a set of objects and an equivalence relation, loosely speaking, the problem asks how to find an accessible way to tell whether two objects are in the same equivalence class. A general approach to this problem is to find a complete set of (geometric, analytic or algebraic) invariants. In the subject of Several Complex Variables and Complex Geometry, a fundamental problem is to classify complex manifolds or more generally, normal complex spaces under the action of biholomorphic transformations. When the normal complex spaces are open and have strongly pseudo-convex boundary, by the Fefferman-Bochner theorem, one needs only to classify the corresponding boundary strongly pseudoconvex CR manifolds under the application of CR diffeomorphisms. The celebrated Chern-Moser theory is a theory which gives two different constructions of a complete set of invariants for such a classification problem. Among various aspects of the Chern-Moser theory (especially the geometric aspect of the theory), the Chern-Moser-Weyl tensor plays a key role. However, this trace-free tensor is defined in a very complicated manner. This makes it hard to apply in the applications. The majority of first several sections in this article surveys some work done in papers of Chern-Moser [3], Huang-Zhang [14], Huang-Zaitsev [13]. Here, we give a simple and more accessible account on the Chern-Moser-Weyl tensor. We also make an immediate application of the monotonicity property for this tensor to the study of CR embedding problem for the positive signature case.