Geodesic
An infinitesimal Rattray theorem
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 148-149 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {V. V. Makeev},
     title = {An infinitesimal {Rattray} theorem},
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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/

[1] R. Rattray, “An antipodal point, orthogonal point theorem”, Ann. of Math., 60 (1954), 502–512 | DOI | MR | Zbl