Weakly complex inner symmetric spaces

TL;DR

This paper characterizes when the tangent bundle of an inner symmetric space of compact type is weakly complex, showing it occurs precisely for products of even-dimensional spheres or Hermitian symmetric spaces.

Contribution

It provides a complete classification of inner symmetric spaces with weakly complex tangent bundles, linking geometric structure to product decompositions.

Findings

Tangent bundle is weakly complex iff the space is a product of spheres or Hermitian symmetric spaces.

Characterizes the geometric conditions for weakly complex tangent bundles.

Connects weakly complex structures to specific symmetric space decompositions.

Abstract

We prove that the tangent bundle of an inner symmetric space $M$ of compact type is weakly complex if and only if $M$ is a Riemannian product $M_1\times...\times M_k$, each $M_i$ being an even-dimensional round sphere or Hermitian symmetric.