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CURVATURE AND COMPLEX SINGULARITIES 509
(iii) We will conclude with some observations about curvature integrals in
the real and complex cases. In real differential geometry one generally encounters
two types of curvature integrals, illustrated for a Riemannian surface by
the two expressions
(5.28) Is KdA, .ts IK]dA
In general the first type includes characteristic classes and Weyl’s coefficients
(1.10), and reflects intrinsic metric properties of the manifold which may even
be of a topological nature. The second type usually describes extrinsic properties
of the manifold in Euclidean space. For a complex manifold M, C
because of sign properties such as
(_ 1)kcc(t) / n-k _> 0
the distinction between the two types of curvature integrals illustrated in (5.28)
seems to disappear, and the situation may be said either to be simpler or more
rigid, depending on one’s viewpoint.
Finally, along similar lines it would seem interesting to try and draw conclusions
on the curvature of real algebraic curves acquiring a singularity. For
example, one might examine the real curves Ct,IR IR C where Ct c is
defined by f(x, y) with f(x, y) being a weighted homogeneous polynomial
having real coefficients. Superficial considerations suggest that the curvature of
Ct,R in IR tends to , while on the Riemann surface Ct the geodesic curvature
of the (purely imaginary) vanishing cycles remains bounded.
(iv) This paragraph is an afterthought. Upon reviewing the preceeding discussion
about "topologically invariant curvature integrals associated to a singularity"
it seems to me that some clarification is desirable.
To begin with let us consider the pedogogical question of how one might best
prove the Gauss-Bonnet theorem
(5.29) ct-e l KdM x(M)
in a course on differential geometry. Chern’s intrinsic proof (cf. footnote (8) in
section 1) is probably the quickest but in my experience leaves some mystery
as to the origin of his formulas. The Allendoerfer-Weil proof (cf. footnote (5) in
section 1) is intuitively appealing and explains the origin of the formula, but
relies on either the Nash embedding theorem or an unpleasant construction to
show that the left hand side of (5.29) is independent of the metric. Finally, the
proof by characteristic classes, interpreting the Gauss-Bonnet integrand as a
de Rham representative of the Euler class of the tangent bundle, is conceptually
satisfying but involves establishing a fairly elaborate machine. The following
proof, in four steps, represents a compromise"
a) Prove (5.29) for oriented real hypersurfaces using the Hopf theorem, as