THE FIXED POINT INDEX OF THE POINCAR ´ E TRANSLATION OPERATOR ON DIFFERENTIABLE MANIFOLDS MASSIMO FURI, MARIA PATRIZIA PERA, AND MARCO SPADINI 1. Introduction The fixed point index of the Poincar´ e translation operator associated to an or- dinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclid- ean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincar´ e translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation op- erator associated with non-autonomous equations will be deduced as consequences. We emphasize that other results, unrelated to our approach, but still involving the fixed point index of the Poincar´ e translation operator have been successfully exploited, for instance, by Srzednicki (see e.g. [Srz2, Srz3]). The chapter is organized as follows. In Section 2 we recall first some elements of calculus on finite dimensional manifolds. Then, in this context, we introduce the notions of Fixed Point Index of a map and of Degree of a tangent vector field. In particular, we show a simple axiomatic approach to the fixed point index theory for maps on a manifold based on just three axioms (see Theorem 2.23 below). In the same section we discuss some basics regarding first order differential equations on differentiable manifolds. In Section 3, we consider the flow on a manifold M induced by an autonomous differential equation of the form ˙ x = g(x), where g is a vector field tangent to M . Then, in the spirit of an earlier result by Krasnosel’skii [Kra], we provide a relation, valid for sufficiently short times, between the degree of a tangent vector field and the fixed point index of its associated flow. From this result, we deduce a formula (see Theorem 3.8 below) for the computation, in terms of the degree of g, of the fixed point index of the flow operator associated with the above equation. This formula is valid whenever the fixed point index of the flow is well defined (and not merely for “sufficiently short” times). The idea behind the proof stems, besides the quoted result by Krasnosel’skii, from more 1