  In this paper we discuss the index problem for geometric differential
operators (Spin-Dirac operator, Gau{\ss}-Bonnet operator, Signature operator)
on manifolds with metric horns. On singular manifolds these operators in
general do not have unique closed extensions. But there always exist two
extremal extensions $D_{min}$ and $D_{max}$. We describe the quotient ${\cal
D}(D_{max}) / {\cal D}(D_{min})$ explicitely in geometric resp. topologic terms
of the base manifolds of the metric horns. We derive index formulas for the
Spin-Dirac and Gau{\ss}-Bonnet operator. For the Signature operator we present
a partial result.
  The first version of this paper was completed August 1995 at the University
of Augsburg.
