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Symmetric spaces as adjoint orbits and their geometries

Leonardo F. Cavenaghi, Carolina Garcia, Lino Grama, Luiz San Martin

Abstract

We realize specific classical symmetric spaces, like the semi-Kähler symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag manifolds. These realizations enable us to describe these cotangent bundles' geodesics and Lagrangian submanifolds. As a final application, we present the first examples of vector bundles over simply connected manifolds with nonnegative curvature that cannot accommodate metrics with nonnegative sectional curvature, even though their associated unit sphere bundles can indeed accommodate such metrics. Our examples are derived from explicit bundle constructions over symmetric flag spaces.

Symmetric spaces as adjoint orbits and their geometries

Abstract

We realize specific classical symmetric spaces, like the semi-Kähler symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag manifolds. These realizations enable us to describe these cotangent bundles' geodesics and Lagrangian submanifolds. As a final application, we present the first examples of vector bundles over simply connected manifolds with nonnegative curvature that cannot accommodate metrics with nonnegative sectional curvature, even though their associated unit sphere bundles can indeed accommodate such metrics. Our examples are derived from explicit bundle constructions over symmetric flag spaces.
Paper Structure (16 sections, 22 theorems, 193 equations, 2 figures)

This paper contains 16 sections, 22 theorems, 193 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lie theoretical description of flag manifolds and symmetric spaces
  3. The basics of root systems and parabolic Lie algebras
  4. Flag Symmetric Spaces and their algebraic aspects
  5. The Cotangent bundle of a symmetric flag space
  6. Characterization of some symmetric spaces and flag manifolds
  7. Symmetric spaces on the adjoint orbit
  8. The dual symmetric space to a symmetric flag space
  9. Symplectic forms and Lagrangian submanifolds
  10. On the geometry of symmetric spaces and associated bundles: geodesics and curvature
  11. Geodesics on symmetric flag spaces and their cotangent spaces
  12. Orthogonal symmetric Lie algebras and conditions to nonnegative curvature on bundles
  13. Explicit descriptions of symmetric flag spaces
  14. Case $A_l$
  15. Case $C_l$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(G,H,\sigma)$ be a symmetric space with $\mathrm{Ad}_G(\mathfrak{h})$ compact and let $(\mathfrak{g},\mathfrak{h},\sigma)$ be its orthogonal symmetric Lie algebra. Take any $G$-invariant Riemannian metric on $G/H$. Then,

Figures (2)

  • Figure 1: Real adjoint orbit.
  • Figure 2: Fibers of $T^*\mathrm{S^1}$ and the submanifold $\mathcal{S}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1
  • Proposition 2.1
  • ...and 25 more