Geodesic
Canonical representation of tangent vectors of Grassmannians
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 147-158

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The structure of the tangent bundle of the real Grassmann manifold $G^+_{p,n}$ under the Plücker embedding (in the exterior algebra of the initial Euclidean space) is studied. Explicit expressions for the relation between decompositions of a tangent vector with respect to different bases of the tangent space are obtained, and a constructive method yielding the canonical (= simplest) decomposition is presented. 
@article{ZNSL_2005_329_a11,
     author = {M. Yu. Nikanorova},
     title = {Canonical representation of tangent vectors of {Grassmannians}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {147--158},
     publisher = {mathdoc},
     volume = {329},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a11/}
}
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M. Yu. Nikanorova. Canonical representation of tangent vectors of Grassmannians. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 147-158. http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a11/