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Degree Theory of Immersed Hypersurfaces

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 $-χ(M)$, where $χ(M)$ is the Euler characteristic of $M$.