Geometry of the real Grassmannian manifolds. Parts I, II
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 2, Tome 246 (1997), pp. 84-107
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The properties of the exterior algebra $\Lambda(\mathbb R^n)$ studied in the paper are related to the Euclidean structure in this algebra induced by the scalar product in $\mathbb R^n$. A geometric interpretation of the interior multiplication for decomposable polyvectors is given. The Cartan criterion of decomposability for the polyvectors is formulated in a coordinateless form. The Pluccer model of the real Grassmannian manifold is realized as a submanifold of the Euclidean space $\Lambda(\mathbb R^n)$, and the isometry of this submanifold onto the classical Grassmannian manifold with $SO(n)$-invariant metric is indicated. For the bivectors the canonical decomposition is described.
@article{ZNSL_1997_246_a4,
author = {S. E. Kozlov},
title = {Geometry of the real {Grassmannian} manifolds. {Parts~I,~II}},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {84--107},
year = {1997},
volume = {246},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_246_a4/}
}
S. E. Kozlov. Geometry of the real Grassmannian manifolds. Parts I, II. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 2, Tome 246 (1997), pp. 84-107. http://geodesic.mathdoc.fr/item/ZNSL_1997_246_a4/