On the existence of vector fields with a given set of singular points on a two-dimensional closed oriented manifold.

*(English. Russian original)*Zbl 0830.57017
Ukr. Math. J. 45, No. 12, 1920-1923 (1993); translation from Ukr. Mat. Zh. 45, No. 12, 1706-1709 (1993).

Summary: We study the possibility of constructing locally gradient and arbitrary vector fields with a given set of singular points on a two-dimensional closed oriented manifold. The sum of the indices of the vector field at these points is equal to the Euler characteristic of the manifold.

##### MSC:

57R25 | Vector fields, frame fields in differential topology |

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

##### Keywords:

locally gradient fields; indices of vector field; vector fields; singular points; two-dimensional closed oriented manifold; Euler characteristic
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\textit{E. A. Girik}, Ukr. Math. J. 45, No. 12, 1706--1709 (1993; Zbl 0830.57017); translation from Ukr. Mat. Zh. 45, No. 12, 1706--1709 (1993)

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##### References:

[1] | A. T. Fomenko,Differential Geometry and Topology. Additional Chapters [in Russian], Moscow University, Moscow (1983). · Zbl 0517.53001 |

[2] | A. S. Mishchenko and A. T. Fomenko,A Course of Differential Geometry and Topology [in Russian], Moscow University, Moscow (1980). · Zbl 0524.53001 |

[3] | J. Milnor and A. Wallace,Differential Topology [Russian translation], Mir, Moscow (1972). |

[4] | H. Poincar?,Selected Works [Russian translation], Nauka, Moscow (1972). |

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