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A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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How to solve optimization problems on manifolds?

To complement Christian Clason's comment: there is usually no need to compute geodesics to optimize over manifolds directly. … For various manifolds of practical interest, computationally inexpensive retractions are known. …
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