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Note on Optimization on Manifold Manifold is locally similar to $\R^n$ flat space, but globally not. Manifold is locally homeomorphic to a Euclidean space of the same dimension. Besides there is Riemann logrithm map that connect the local vector space $T_p$ to the neighbourhood of $p$ on the manifold. Thus, many local optimization algorithm that work on flat $\R^n$ space can work the same on neighborhood of a manifold. The unique thing of working on a manifold is how to transport the local direction information in one neighborhood into another neighborhood.