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 2n is an oriented 2n-dimensional closed manifold with non-positive curvature, then is it true that its Euler number χ(M 2n) satisfies the inequality (−1) nχ(M 2n) ≥ 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 2, g) is an oriented closed Riemmanian surface with curvature secg and the Euler number χ(M 2). It follows from Gauss-Bonnet formula that the total Gauss curvature is equal to the Euler number: M 2 secgdA = 2πχ(M 2). (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 2n, g) of complex dimension n = dimC(M 2n). For such a Kähler manifold (M 2n, g), there is a top-dimensional Chern-form cn(M 2n) = cn(T (1,0) M 2n) which is equal to the Eule
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