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428 PHILLIP A. GRIFFITHS
Introduction
This paper closely parallels the Gergen lectures which I was fortunate to
have the opportunity to deliver at Duke University in March, 1978. My goal in
those talks was to discuss the relationship between curvature and the singularities
of complex-analytic varieties which was initiated in the recent work of
Linda Ness1), N. A’Campo, and R6mi Langevin), and which is somewhat
broadened and extended here. Since these lectures were intended for a general
audience, much of the first two sections of the paper is expository, covering
known material but from a viewpoint intended to serve later needs. Indeed, the
discussion of real differential and integral geometry is not logically necessary
for the eventual results on curvature and singularities, but by thinking through
this more intuitive and familiar material the extension to the complex case
should appear natural, and moreover should permit deeper appreciation of the
special features peculiar to Hermitian differential and integral geometry. In the
remainder of this introduction a somewhat more detailed account of the contents
of this paper will be given.
In the first section there is a discussion of Hermann Weyl’s formula for the
volume of the tube r(M) of radius r around an oriented n-manifold M in IRN.
The result is
(0.1) vol rr(M) m C(n, m, k) tz(M)r +
k=O
where rn N n, the C(n, m, k) are universal constants, and the
Ix(M) C(k, n) It I(nt)dM
are integrals over M of scalar invariants I(R) constructed from the Riemann
curvature tensor R. The point is that the right-hand side of (0.1) is an intrinsic
invariant of the induced Riemannian metric and does not depend on the particular
embedding M C IRN. At one extreme/x0(M) vol (M) (as expected), while
at the other extreme I,(Rt) is zero for n odd and for n even it is the integrand
K dM in the general Gauss-Bonnet theorem
(0.2) Ct It K dM x(M)
for a closed manifold. (z) In fact, it was by using tubes (their boundary, to be
precise) that the general Gauss-Bonnet theorem was first deduced from the
formula (0.2) for oriented hypersurfaces, where K is then the Gauss-Kronecker
curvature. We have recounted this development in lb, partly because it is of
historical interest and illustrates the observation that in geometry intrinsic formulas
are frequently arrived at by extrinsic considerations, and partly because
the Gauss-Bonnet is certainly the most important curvature integral. In c we
show how, via the Gauss mapping, this extrinsic proof ties in with the more
modern version using characteristic classes. The first section concludes with