Geodesic
An infinitesimal Rattray theorem
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 148-149

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Let $X$ be a continuous vector field on the unit Euclidean sphere centered at the origin such that $X(-a)=-X(a)$. It is proved that there is an orthonormal basis in the space such that for any two vectors $a$ and $b$ in the basis we have $X(a)\cdot b+a\cdot X(b)=0$. Bibl. – 1 title. 
@article{ZNSL_2008_353_a13,
     author = {V. V. Makeev},
     title = {An infinitesimal {Rattray} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {148--149},
     publisher = {mathdoc},
     volume = {353},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a13/}
}
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V. V. Makeev. An infinitesimal Rattray theorem. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 148-149. http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a13/