Curvature, integrability, and the six sphere

Gabriella Clemente

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arXiv · 2108.02586

Curvature, integrability, and the six sphere

Gabriella Clemente

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Abstract

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not hermitian. The partial proof is intrinsic in nature, and shows some level of promise when it comes to generalizing the non-complexity of the round $S^6$ result in new directions.

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Gabriella Clemente. 2021-08-04. Curvature, integrability, and the six sphere. https://arxiv.org/abs/2108.02586

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