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A symmetric space is a connected Riemannian manifold in which at every point there exists a global self-isometry whose differential at the given point is minus identity.

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Classification of "homogeneous" submanifolds of ℝⁿ

Such $M$ are called extrinsically homogeneous submanifolds of euclidean space. Reference: for example the book by Berndt-Console-Olmos, Submanifolds and holonomy. The set of all euclidean isometries t …
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