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In this paper, we define a class of new geometric flows on a completeRiemannian manifold. The new flow is related to the generalized (third order)Landau-Lifishitz equation. On the other hand it could be thought of a specialcase of the Schr\"odinger-Airy flow when the target manifold is a K\"ahlermanifold with constant holomorphic sectional curvature. We show the localexistence of the new flow on a complete Riemannian manifold with someassumptions on Ricci tensor. Moreover, if the target manifolds are Einstein orsome certain type of locally symmetric spaces, we obtain the global results.

New Geometric Flows on Riemannian Manifolds and Applications to Schr\"odinger-Airy Flows