The Gauss-Bonnet theorem and the Atiyah-Patodi-Singer functionals for the characteristic classes of foliations

WE PROVE in this article some formulae of Gauss-Bonnet kind for the characteristic classes of foliations (see [2], [S]). Namely, fixing a Riemannian metric on a manifold with a foliation allows one to determine explicitly for each of these characteristic classes a representing differential form (see, for example, [4]). If a domain X with a piecewise smooth boundary transversal to the foliation is given in the manifold, then the integral of such a form over X, corrected by adding the integrals over faces of different dimensions of certain forms depending on the foliation and the metric near 8X, depends only on the induced metric on aX (and, of course, on the foliation on X). By using these formulae one can, in particular, extend the definition of the characteristic classes of foliations to the piecewise smooth case. The general idea of making topological invariants local by means of explicit formulae depending on a metric or another auxiliary structure was suggested by I. M. Gel’fand at the Nice congress[6]. This idea is partially realized here for the characteristic classes of foliations (though the general theory developed in §§2,4 and 5 will possibly have wider applications). Remarkable progress in this direction has been achieved in the recent work of Atiyah-Patodi-Singer [ 11 where a formula of Gauss-Bonnet kind is obtained for the Hirzebruch polynomials (of the Pontryagin forms) and the signature. The essentially new aspect of the Atiyah-Patodi-Singer formula is that its right-hand-side-a quantity depending on the induced metric of the boundary-is not expressible as the integral over the boundary of a form locally determined by the metric. (It measures the degree of the asymmetry of the spectrum of the elliptic operator in the space of forms which takes cp into (-l)‘(d * cp * dcp), where p = (deg cp)/2). This property of the ‘boundary functional’-its ‘non-localizability’ is still valid in our formula. In further comparing the Atiyah-Patodi-Singer formula with ours we should point out an important advantage of the former. Our boundary functional depends on the auxiliary structure on the boundary and on the whole foliation. The corresponding part of the Atiyah-Patodi-Singer formula is divided into two parts: a quantity depending only on the metric properties of the boundary, and a quantity depending only on the topological properties of the whole manifold. We do not see how to obtain such a decomposition of our boundary functional. More detailed discussion of the interrelations between our formula, the Atiyah-Patodi-Singer formula, and the classical Gauss-Bonnet formula will be found in # 4 of 91. The first section is devoted to codimension one foliations; in its plan it is a reduced copy of the remainder of the article. We have tried to write it more concretely and without omitting calculations, especially so because the method of calculation is often of more importance for us than the result. The contents of the remaining sections will be briefly described at the end of 91.