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Lagrangian surfaces of constant angle in the complex Euclidean plane

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Abstract

We investigate some geometric properties of Lagrangian surfaces, in the complex Euclidean plane \(\mathbb {C}^2\), which make a constant angle with respect to a parallel vector field Z. This latter condition means that the angle function, between the tangent planes of the surface and Z, is a constant. The first basic property is that if M is a Lagrangian surface of constant angle with respect to Z, then it also has constant angle with respect to JZ, where J is the standard almost complex structure on \(\mathbb {C}^2\). In particular, we have that its tangent components \(Z^\top \) and \(J Z^\perp \) are two vector fields on M of constant length. When they are linearly dependent we deduce that M should be part of a Lagrangian cylinder. When they are linearly independent, we have a frame on M. We use this frame to investigate the properties of these surfaces. The Gaussian curvature is not necessarily constant zero as in the case of the Lagrangian cylinder. In particular, if the angle between \(Z^\top \) and \(J Z^\perp \) is constant we prove that M is part of a Lagrangian plane. Finally, we investigate these surfaces with a parametrization and we give a system of two PDE’s in two variables that are the equivalent conditions to be a Lagrangian surface of constant angle.

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Acknowledgements

We are very grateful by the revision and corrections by the referee which help us to improve our manuscript.

Funding

The first author was supported with a doctoral scholarship by CONACYT. Both authors were partially supported under the project UNAM-PAPIIT IN117720.

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Correspondence to Gabriel Ruiz-Hernández.

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Aguilar-Suárez, R., Ruiz-Hernández, G. Lagrangian surfaces of constant angle in the complex Euclidean plane. Bol. Soc. Mat. Mex. 28, 70 (2022). https://doi.org/10.1007/s40590-022-00460-5

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  • DOI: https://doi.org/10.1007/s40590-022-00460-5

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