  In this paper we state and prove a higher index theorem for an
odd-dimensional connected spin riemannian manifold $(M,g)$ which is partitioned
by an oriented closed hypersurface $N$. This index theorem generalizes a
theorem due to N. Higson and J. Roe in the context of Hilbert modules. Then we
apply this theorem to prove that if $N$ is area-enlargeable and if there is a
smooth map from $M$ into $N$ such that its restriction to $N$ has non-zero
degree then the the scalar curvature of $g$ cannot be uniformly positive.
