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 ${\mathbb{S}}^{2n+1}$, which admits a Sasakian structure $(\phi ,\xi ,\eta ,g)$ for $n\ge 2$. 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 ${\mathbb{S}}^{2n+1}$ with conformal Maslov form, if its sectional curvature satisfies the pinching $0\le {\sec }_g\le 1$, 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.

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