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J. GE
J. GE
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For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4-pinched curvature). It has totally geodesic embedded $\mathbb CP^{n-1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space.

Edit: If you prefer non-Riemannian Alexandrov space, take the spherical suspension of $\mathbb CP^n$.

For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4-pinched curvature). It has totally geodesic embedded $\mathbb CP^{n-1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space.

For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4-pinched curvature). It has totally geodesic embedded $\mathbb CP^{n-1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space.

Edit: If you prefer non-Riemannian Alexandrov space, take the spherical suspension of $\mathbb CP^n$.

Source Link
J. GE
J. GE
  • 2.6k
  • 14
  • 26

For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4-pinched curvature). It has totally geodesic embedded $\mathbb CP^{n-1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space.