  We develop a degree theory for compact immersed hypersurfaces of prescribed
$K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$
is any elliptic curvature function. We apply this theory to count the
(algebraic) number of immersed hyperspheres in various cases: where $K$ is mean
curvature; extrinsic curvature and special Lagrangian curvature, and we show
that in all these cases, this number is equal to $-\chi(M)$, where $\chi(M)$ is
the Euler characteristic of $M$.
