Symmetric spaces as adjoint orbits and their geometries

TL;DR

This paper represents classical symmetric spaces as cotangent bundles of symmetric flag manifolds, enabling detailed geometric analysis and providing novel examples of vector bundles with nonnegative curvature properties.

Contribution

It introduces a new realization of symmetric spaces as cotangent bundles, facilitating the study of their geodesics and Lagrangian submanifolds, and constructs the first examples of certain curvature-restricted vector bundles.

Findings

Realized symmetric spaces as cotangent bundles of flag manifolds

Described geodesics and Lagrangian submanifolds in these bundles

Constructed vector bundles with nonnegative curvature properties

Abstract

We realize specific classical symmetric spaces, like the semi-K\"ahler 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.