Indices of Vector Fields on Real Analytic Varieties

Abstract

In the previous chapters we focused on indices of vector fields on complex analytic varieties. The real analytic setting also has its own interest, and that is the subject of this chapter. The following presentation follows the discussion by M. Aguilar, J. Seade and A. Verjovsky in [6] (see also [49]). We describe indices analogous to the GSV and Schwartz indices for vector fields on real analytic singular varieties. In this setting the GSV index is an integer if the singular variety V is odd-dimensional, but it is defined only modulo 2 if the dimension of V is even.

The Schwartz and the GSV indices are defined, respectively, in Sects. 1 and 2; there we show that the Schwartz index classifies the homotopy classes of vector fields near an isolated singularity. Section 3 provides a geometric interpretation of the GSV index in the real analytic setting.

The information we get is related to previous work by M. Kervaire about the curvatura integra of manifolds, and this is the subject we explore in Sect. 4. Finally, in Sect. 5 we look at the relation of these indices with other invariants of real analytic singularity germs studied previously by C. T. C. Wall and others. This yields to an extension of the concept of Milnor number for real analytic map-germs with isolated singularities which may not be algebraically isolated.

We note that there are some related works such as [9, 10, 49]. Since in this chapter we consider only real analytic varieties and functions, for simplicity, we will denote the dimensions here by m, n... instead of m’, n’…, as in the rest of the book.