Geodesic
A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in $S^3$
Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 92-105

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We study a version of the Gauss map $g\ :M^2\to S^2$ for a surface $M^2$ immersed in $S^3$ and prove an analog of the Ruh–Vilms theorem which states that this map is harmonic if $M^2$ has a constant mean curvature. As a corollary, we conclude that an embedded flat torus $T^2$ with constant mean curvature is a spherical Delonay surface. 
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     author = {L. A. Masal'tsev},
     title = {A {Version} of the {Ruh--Vilms} {Theorem} for {Surfaces} of {Constant} {Mean} {Curvature} in $S^3$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {92--105},
     publisher = {mathdoc},
     volume = {73},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_1_a7/}
}
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L. A. Masal'tsev. A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in $S^3$. Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 92-105. http://geodesic.mathdoc.fr/item/MZM_2003_73_1_a7/