We investigate geometric properties of the (Sato–Segal–Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers–Størmer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations.We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds. We revisit the calculation of Calabi’s diastasis function given by Spera and Valli again in the context of C∗-algebras, using the τ-function to give an expression of the diastasis on the infinitedimensional Grassmannian; this expression can be applied to the image of the Krichever map to give a proof of Weil’s reciprocity based on the fact that the distance of two points on the Grassmannian is symmetric. Another application is the fact that each (isometric) automorphism of the Grassmannian is induced by a projective transformation in the Pl¨ucker embedding.
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