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A note on precotangent bundles: the example of Grassmannians

Tomasz Goliński

TL;DR

The work addresses the existence of precotangent (predual) bundles to Grassmannians in Banach geometry, focusing on Grassmannians of reflexive spaces and $p$-restricted Grassmannians. It constructs $T_*\mathrm{Gr}$ as a subbundle of $T^*\mathrm{Gr}$ with fibers modeled on the appropriate preduals (trace-class or $L^0$), and verifies global compatibility via chart transitions. Key results include the explicit existence of $T_*\mathrm{Gr}(\mathcal H)$ for Hilbert spaces, the $p=1$ case for polarized spaces, and a general existence theorem for $\mathrm{Gr}(\mathbb E)$ when $\mathbb E$ is reflexive, with extensions to graded variants like $\mathrm{Gr}_k$ and $\mathbb{CP}(\mathbb E)$. These constructions underpin Banach-Lie-Poisson geometry on Grassmannians and related bundles.

Abstract

We prove the existence of the bundle predual to the tangent bundle (called precotangent bundle) for Grassmannians of reflexive Banach spaces and $p$-restricted Grassmannians of the polarized Hilbert space.

A note on precotangent bundles: the example of Grassmannians

TL;DR

The work addresses the existence of precotangent (predual) bundles to Grassmannians in Banach geometry, focusing on Grassmannians of reflexive spaces and -restricted Grassmannians. It constructs as a subbundle of with fibers modeled on the appropriate preduals (trace-class or ), and verifies global compatibility via chart transitions. Key results include the explicit existence of for Hilbert spaces, the case for polarized spaces, and a general existence theorem for when is reflexive, with extensions to graded variants like and . These constructions underpin Banach-Lie-Poisson geometry on Grassmannians and related bundles.

Abstract

We prove the existence of the bundle predual to the tangent bundle (called precotangent bundle) for Grassmannians of reflexive Banach spaces and -restricted Grassmannians of the polarized Hilbert space.

Paper Structure

This paper contains 5 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Initial glance: Grassmannian of a Banach space
  3. Grassmannian of the Hilbert space
  4. $p$-restricted Grassmannians
  5. Grassmannian of a reflexive Banach space

Key Result

Theorem 2

The precotangent bundle $T_*\mathop{\mathrm{Gr}}\nolimits(\mathcal{H})$ exists and locally it is defined as

Theorems & Definitions (9)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Remark 6