HyperKahler Contact Distributions
Abstract
Let $(\varphi_\alpha,\xi_\alpha,g)$ for $\alpha=1,2$, and $3$ be a contact metric $3$-structure on the manifold $M^{4n+3}$. We show that the $3$-contact distribution of this structure admits a HyperKahler structure whenever $(M^{4n+3},\varphi_\alpha,\xi_\alpha,g)$ is a $3$-Sasakian manifold. In this case, we call it HyperKahler contact distribution. To analyze the curvature properties of this distribution, we define a special metric connection that is completely determined by the HyperKahler contact distribution. We prove that the $3$-Sasakian manifold is of constant $\varphi_{\alpha}$-sectional curvatures if and only if its HyperKahler contact distribution has constant holomorphic sectional curvatures.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.05348
- arXiv:
- arXiv:2109.05348
- Bibcode:
- 2021arXiv210905348A
- Keywords:
-
- Mathematics - Differential Geometry
- E-Print:
- arXiv admin note: substantial text overlap with arXiv:1204.3407