CURVATURE IDENTITIES DERIVED FROM AN INTEGRAL FORMULA FOR THE FIRST CHERN NUMBER

We establish an integral formula for the first Chern number of a compact almost Hermitian surface and derive curvature identities from the integral formula. Further, we provide some results as applications of the identities. In (1), Berger derived a curvature identity on a 4-dimensional compact ori- ented Riemannian manifold from the generalized Gauss-Bonnet formula based on the fundamental fact that the Euler number is a topological invariant. Kuz'mina (11) and subsequently Labbi (12) extended Berger's result to any even-dimensional Riemannian manifold. Especially, Labbi showed that the obtained curvature identities hold without the compactness assumption. Euh, Park and Sekigawa (3) gave a direct proof for Labbi's result in the 4-dimensional case and some applications of the curvature identity (4, 5). We refer (6) for the universality of the curvature identities in the Riemannian setting and we refer (7) for the universality of the curvature identities in the pseudo-Riemannain setting. Motivated by the above observation, it is also worthwhile to study similar topics for the Riemannian manifolds equipped with some additional geometric structures such as almost complex structure, almost contact structure and so on. In this paper, we will focus on the following question: Question A. What can we deduce from the integral formula for the Chern number of compact almost Hermitian surfaces? We first establish an integral formula for the Chern number of a compact almost Hermitian surface. We regard the obtained integral formula as a func- tional on the space of all almost Hermitian structures on a compact complex