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We consider the generalized evolution of compact level sets by functions oftheir normal vectors and second fundamental forms on a Riemannian manifold M.The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever usolves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfyinga geometric condition. We show existence and uniqueness of viscosity solutionsto this equation under the assumptions that M has nonnegative curvature, F iscontinuous off Du=0, (degenerate) elliptic, and locally invariant by paralleltranslation. We then prove that this approach is geometrically consistent,hence it allows to define a generalized evolution of level sets by verygeneral, singular functions of their curvatures. For instance, theseassumptions on F are satisfied when F is given by the evolutions of level setsby their mean curvature (even in arbitrary codimension) or by their positiveGaussian curvature. We also prove that the generalized evolution is consistentwith the classical motion by the corresponding function of the curvature,whenever the latter exists. When M is not of nonnegative curvature, the sameresults hold if one additionally requires that F is uniformly continuous withrespect to D^2 u. Finally we give some counterexamples showing that severalwell known properties of the evolutions in $\mathbb{R}^{n}$ are no longer truewhen M has negative sectional curvature.

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds