Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2017
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1712.04478 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910294770450432 |
|---|---|
| author | Lee, Ruenn-Huah Loo, Tee-How |
| author_facet | Lee, Ruenn-Huah Loo, Tee-How |
| contents | Let $M$ be a real hypersurface in complex Grassmannians of rank two. Denote by $\mathfrak J$ the quaternionic Kähler structure of the ambient space, $TM^\perp$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$. The real hypersurface $M$ is said to be $\mathfrak D^\perp$-invariant if $\mathfrak D^\perp$ is invariant under the shape operator of $M$. We showed that if $M$ is $\mathfrak D^\perp$-invariant, then $M$ is Hopf. This improves the results of Berndt and Suh in [{Int. J. Math.} \textbf{23}(2012) 1250103] and [{Monatsh. Math.} \textbf{127}(1999), 1--14]. We also classified $\mathfrak D^\perp$ real hypersurface in complex Grassmannians of rank two with constant principal curvatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_04478 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | $\mathfrak D^\perp$-invariant real hypersurfaces in complex Grassmannians of rank two Lee, Ruenn-Huah Loo, Tee-How Differential Geometry Let $M$ be a real hypersurface in complex Grassmannians of rank two. Denote by $\mathfrak J$ the quaternionic Kähler structure of the ambient space, $TM^\perp$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$. The real hypersurface $M$ is said to be $\mathfrak D^\perp$-invariant if $\mathfrak D^\perp$ is invariant under the shape operator of $M$. We showed that if $M$ is $\mathfrak D^\perp$-invariant, then $M$ is Hopf. This improves the results of Berndt and Suh in [{Int. J. Math.} \textbf{23}(2012) 1250103] and [{Monatsh. Math.} \textbf{127}(1999), 1--14]. We also classified $\mathfrak D^\perp$ real hypersurface in complex Grassmannians of rank two with constant principal curvatures. |
| title | $\mathfrak D^\perp$-invariant real hypersurfaces in complex Grassmannians of rank two |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1712.04478 |