The Exponential Map and the Index Density

Abstract

In this chapter, we will give another proof of Theorem 4.1 for a Dirac operator D, independent of the one in Chapter 4. This proof generalizes easily to obtain the fixed point formula for the equivariant index of a group of isometries, as we will see in the next chapter. A feature of the proof is that it gives an explanation for the striking similarity between the Â-genus and the Jacobian of the exponential map on a Lie group, both of which involve the j-function

$$ j\left( X \right) = \frac{{\sinh \left( {X/2} \right)}} {{X/2}} $$

.