By Michael Spivak
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Extra info for A Comprehensive Introduction to Differential Geometry Volume 2, Third Edition
Clearly it is equal to the Grassmannian of (l + 1)-dimensional complex linear subspaces in Cn+1 . 4) l where dE is the Haar measure on C G normalized in some way (we do not care about normalization constants). 5) where c = 0 is a normalizing constant depending on normalizations of Haar measures and l, m, n. Let us give a heuristic proof of this equality. 8, we observe that φl (K) = 1lE dE (K), CG l where 1lE is considered as a generalized valuation. 14. Since for generic projective subspaces E and F their intersection E ∩ F is a projective subspace of dimension l + m − n for l + m ≥ n and empty otherwise, it follows that CG l× 1lM dM = c · φl+m−n .
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A Comprehensive Introduction to Differential Geometry Volume 2, Third Edition by Michael Spivak