In this paper, we study the family index of a family of spin
manifolds. In particular, we discuss to what extent the real index (of
the Dirac operator of the real spinor bundle if the fiber dimension is
divisible by 8) which can be defined in this case contains extra
information over the complex index (the index of its
complexification). We study this question under the additional
assumption that the complex index vanishes on the k–skeleton of
B. In this case, we define new analytical invariants c^k in
Hk-1(B;R/Z), certain secondary invariants.
We give interesting nontrivial examples. We then describe this
invariant in terms of known topological characteristic classes.