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- W1964037703 abstract "Let $$M$$ and $$N$$ be two connected smooth manifolds, where $$M$$ is compact and oriented and $$N$$ is Riemannian. Let $$mathcal {E}$$ be the Fréchet manifold of all embeddings of $$M$$ in $$N$$ , endowed with the canonical weak Riemannian metric. Let $$sim $$ be the equivalence relation on $$mathcal {E}$$ defined by $$fsim g$$ if and only if $$f=gcirc phi $$ for some orientation preserving diffeomorphism $$phi $$ of $$M$$ . The Fréchet manifold $$mathcal {S}= mathcal {E}/_{sim }$$ of equivalence classes, which may be thought of as the set of submanifolds of $$N$$ diffeomorphic to $$M$$ and is called the nonlinear Grassmannian (or Chow manifold) of $$N$$ of type $$M$$ , inherits from $$ mathcal {E}$$ a weak Riemannian structure. We consider the following particular case: $$N$$ is a compact irreducible symmetric space and $$M$$ is a reflective submanifold of $$N$$ (that is, a connected component of the set of fixed points of an involutive isometry of $$ N$$ ). Let $$mathcal {C}$$ be the set of submanifolds of $$N$$ which are congruent to $$M$$ . We prove that the natural inclusion of $$mathcal {C}$$ in $$mathcal {S}$$ is totally geodesic." @default.
- W1964037703 created "2016-06-24" @default.
- W1964037703 creator A5011944385 @default.
- W1964037703 date "2014-05-31" @default.
- W1964037703 modified "2023-10-03" @default.
- W1964037703 title "Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space" @default.
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- W1964037703 doi "https://doi.org/10.1007/s00605-014-0642-2" @default.
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