Gauss-bonnet-chern Formulae and Related Topics for Curved Riemannian Manifolds

In this paper, we survey recent results on Gauss-Bonnet-Chern formulae and related issues for closed Riemannian manifolds with variable curvature. Among other things, we address the following problem: “if M is an oriented 2n-dimensional closed manifold with non-positive curvature, then is it true that its Euler number χ(M) satisfies the inequality (−1)χ(M) ≥ 0?” We will present some partial answer to this question in Kähler case. In addition, we discuss some related results to characterize a curved manifolds via various geometric invariants, along the line of Professor Chern. §1. Curvature, Gauss-Bonnet-Chern formulae and Euler numbers. One of important contributions of Professor Chern to global differential geometry is the celebrated Gauss-Bonnet-Chern formula. Let us recall the classical Gauss-Bonnet formula for closed surfaces. Suppose that (M, g) is an oriented closed Riemmanian surface with curvature secg and the Euler number χ(M ). It follows from GaussBonnet formula that the total Gauss curvature is equal to the Euler number: ∫ M2 secgdA = 2πχ(M ). (1.1) Professor Chern was able to extend Gauss-Bonnet formula to higher dimensional manifolds. For instance, let us consider an oriented closed Kähler manifold (M, g) of complex dimension n = dimC(M ). For such a Kähler manifold (M, g), there is a top-dimensional Chern-form cn(M ) = cn(T M) which is equal to the Euler *The first author was supported in part by an NSF grant and the second author was partially supported by a Chinese NSF grant. Typeset by AMS-TEX 1 class e(M), (see [BoT82] page 273, [MS74] page 155). When (M, g) is a Kähler closed manifold, the Gauss-Bonnet-Chern formula reads as ∫