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.
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Acknowledgements
The São Paulo Research Foundation (FAPESP) supports L. F. C. grants 2022/09603-9, 2023/14316-1, and partially supports L. G. grants 2023/13131-8, 2021/04065-6, and L. S. M, grant 2018/13481-0. C. G. was supported by CAPES Ph.D. Scholarship. The authors gladly acknowledge the anonymous referees for the careful read, leading to useful suggestions that improved the quality of the paper.
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Cavenaghi, L.F., Garcia, C., Grama, L. et al. Symmetric spaces as adjoint orbits and their geometries. Rev Mat Complut (2024). https://doi.org/10.1007/s13163-024-00486-5
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DOI: https://doi.org/10.1007/s13163-024-00486-5