Tangent Spaces to Grassmannians
We have seen that the Grassmannian 𝔾(k, n) is a smooth variety of dimension (k + 1) (n - k). This follows initially from our explicit description of the covering of 𝔾 (k, n) by open sets U Λ ≅ 𝔸(k+1)(n-k), though we could also deduce this from the fact that it is a homogeneous space for the algebraic group PGL n+1 K. The Zariski tangent spaces to G are thus all vector spaces of this dimension. For many reasons, however, it is important to have a more intrinsic description of the space T Λ(𝔾;) in terms of the linear algebra of Λ ⊂ K n+1. We will derive such an expression here and then use it to describe the tangent spaces of the various varieties constructed in Part I with the use of the Grassmannians.
KeywordsProjective Space Tangent Space Tangent Vector Tangent Plane Smooth Point
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