Poincare’s vector field theorem

The Poincaré vector field theorem (also known as the Poincaré – Hopf theorem and the index theorem ) is a classical theorem of differential topology and theory of dynamical systems; generalization and refinement of the hedgehog combing theorem .

From it, in particular, it follows that on a two-dimensional sphere there does not exist a smooth vector field without singular points, and on a two-dimensional torus it can exist.

Content

1 Formulation
2 History
3 Variations and generalizations
4 notes
5 Literature

Wording

Let on a smooth closed manifold $M$ ${\ displaystyle M}$ smooth vector field defined $V$ ${\ displaystyle V}$ having a finite number of isolated singular points $A_{1},A_{2},\dots ,A_{n}$ ${\ displaystyle A_ {1}, A_ {2}, \ dots, A_ {n}}$ . Then

\sum _{i=1}^{n}J_{A_{i}}(V)=\chi (M),

{\ displaystyle \ sum _ {i = 1} ^ {n} J_ {A_ {i}} (V) = \ chi (M),}

here $J_{A_{i}}(V)$ ${\ displaystyle J_ {A_ {i}} (V)}$ - point index $A_{i}$ ${\ displaystyle A_ {i}}$ relative to the field $V$ ${\ displaystyle V}$ and number $\chi (M)$ ${\ displaystyle \ chi (M)}$ - Euler characteristic of manifold $M$ ${\ displaystyle M}$ .

History

For the case of two-dimensional manifolds, the theorem was proved by Poincare in 1885. For manifolds of arbitrary dimension, the result was obtained by Hopf in 1926 ^[1] .

Variations and generalizations

Similar theorems were proved for vector fields with nonisolated singular points and for manifolds with singularities ^[2] ^[3] .

Notes

↑ A two-dimensional version of this theorem was proved by Poincare in 1885. The complete theorem was proved by Hopf in 1926, following the partial results of Brauer and Hadamard . // Milnor J., Wallace A. Differential topology. Beginner course. M: Mir, 1972 (p. 223).
↑ Jean-Paul Brasselet, José Seade, Tatsuo Suwa . Vector fields on Singular Varieties. Springer, 2009
↑ Pavao Mardešić . Index of singularities of real vector fields on singular hypersurfaces. Journal of Singularities, vol 9 (2014), 111-121

Literature

Milnor J., Wallace A. , Differential Topology. Beginner course. M: World, 1972.
Arnold V.I. , Ordinary differential equations. Any edition.

[1] A two-dimensional version of this theorem was proved by Poincare in 1885. The complete theorem was proved by Hopf in 1926, following the partial results of Brauer and Hadamard . // Milnor J., Wallace A. Differential topology. Beginner course. M: Mir, 1972 (p. 223).

[2] Jean-Paul Brasselet, José Seade, Tatsuo Suwa . Vector fields on Singular Varieties. Springer, 2009

[3] Pavao Mardešić . Index of singularities of real vector fields on singular hypersurfaces. Journal of Singularities, vol 9 (2014), 111-121