Legendrian submanifolds in the unit sphere with conformal Maslov form and constant sectional curvature
Abstract
This paper is concerned with the study on Legendrian submanifolds with conformal Maslov form in the unit sphere 𝕊2n+1, which admits a Sasakian structure $(φ,ξ,η,g)$ for n≥2. As the main result, we classify such submanifolds with constant sectional curvature, motivated by the classification result of the minimal Legendrian submanifolds with constant sectional curvature. Moreover, we prove that, for a closed Legendrian submanifold Mn in 𝕊2n+1 with conformal Maslov form, if its sectional curvature satisfies the pinching 0≤secg≤1, then either Mn is the totally geodesic Legendrian sphere with secg=1, or Mn is a closed embedded weighted Clifford torus with secg=0. This extends the corresponding pinching theorem of Dillen--Vrancken (J Math Pures Appl 69:85--93 1990) from minimal Legendrian submanifolds to the conformal Maslov class under the same curvature bounds.
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BibTeX
@misc{luo2026legendrian,
title = {{Legendrian submanifolds in the unit sphere with conformal Maslov form and constant sectional curvature}},
author = {Yong Luo and Cheng Xing},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14021},
eprint = {2609.14021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14021},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Legendrian submanifolds in the unit sphere with conformal Maslov form and constant sectional curvature
AU - Yong Luo
AU - Cheng Xing
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14021
UR - https://arxiv.org/abs/2609.14021
AB - This paper is concerned with the study on Legendrian submanifolds with conformal Maslov form in the unit sphere 𝕊^2n+1, which admits a Sasakian structure $(φ,ξ,η,g)$ for n≥2. As the main result, we classify such submanifolds with constant sectional curvature, motivated by the classification result of the minimal Legendrian submanifolds with constant sectional curvature. Moreover, we prove that, for a closed Legendrian submanifold M^n in 𝕊^2n+1 with conformal Maslov form, if its sectional curvature satisfies the pinching 0≤sec_g≤1, then either M^n is the totally geodesic Legendrian sphere with sec_g=1, or M^n is a closed embedded weighted Clifford torus with sec_g=0. This extends the corresponding pinching theorem of Dillen--Vrancken (J Math Pures Appl 69:85--93 1990) from minimal Legendrian submanifolds to the conformal Maslov class under the same curvature bounds.
ER -