Bott Periodicity for Inclusions of Symmetric Spaces

When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unitary group. Standard inclusions between other classical Riemannian symmetric spaces are considered as well. An application to homotopy theory is also discussed.

2010 Mathematics Subject Classification: Primary 53C35; Secondary 55R45, 53C40.

Keywords and Phrases: Symmetric spaces, shortest geodesics, reflective submanifolds, Bott periodicity.

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