Secondary Chern-euler Class for General Submanifold

We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give conceptual proofs of a classical result of Chern. The objective of this paper is to define, study and use the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. We give the definition in Definition 1 in Section 1. In Section 2, we study cohomologically the class and its relation with several other natural homology and cohomology classes. The cases when the codimension of the submanifold is one or greater than one are different, and we consider both cases. In Section 3, we use the secondary Chern-Euler class to define index for a vector field with non-isolated singularities on a submanifold in Definition 2. To this end, we develop the notion of blow-up of the submanifold along the vector field. We then obtain three formulas in Theorem 2 to compute the index. Our studies, in particular, give three conceptual proofs of a classical result of Chern [Che45, (20)] concerning the paring of the secondary Chern-Euler class with the normal sphere bundle of the submanifold. Two of the proofs are given in Section 2, while the third in Section 3. 1. Secondary Chern-Euler class for a general submanifold Let X be a connected oriented compact Riemannian manifold of dimension n. (Throughout the paper, n = dimX .) The Gauss-Bonnet theorem (see, e.g., [Che44, (9)]) asserts that