Geodesic
Isometric equivalence of directions in Riemannian symmetric spaces
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 56-57

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It is proved that there exists an isometry of the real Graccmann manifold that puts into coincidence two one-dimensional tangent directions of a totally geodesic 2-surface with nonzero Gauss curvature. 
@article{ZNSL_2005_329_a3,
     author = {S. E. Kozlov},
     title = {Isometric equivalence of directions in {Riemannian} symmetric spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {56--57},
     publisher = {mathdoc},
     volume = {329},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a3/}
}
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S. E. Kozlov. Isometric equivalence of directions in Riemannian symmetric spaces. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 56-57. http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a3/