No CrossRef data available.
Article contents
On conformally flat minimal Legendrian submanifolds in the unit sphere
Published online by Cambridge University Press: 10 May 2024
Abstract
This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb {S}^{2n+1}$
admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$
for $n\ge 3$
, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$
is semi-parallel, which is introduced as a natural extension of $C$
-parallel second fundamental form $h$
. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1027.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1028.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1029.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1030.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1031.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1032.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1033.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1034.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1035.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1036.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1037.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1038.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1039.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1040.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1041.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1042.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1043.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1044.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1045.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509204823580-0604:S030821052400057X:S030821052400057X_inline1046.png?pub-status=live)