Geometry of real grassmann manifolds. Parts I, II

The paper deals with the properties of the exterior algebra ℝ(Λⁿ) related to the Euclidean structure on ℝ(Λⁿ) induced by the scalar product in ℝ(Λⁿ). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λⁿ), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.