176 V.M. Buchstaber
An (autonomous)dynamic systemon a manifoldMis a smooth vector field
ξonM. In terms of the local coordinates{xjα}onM, a dynamical systemξ
gives rise to the system of (autonomous)ordinary differential equations
x ̇jα=ξj(x^1 α,...,xnα).The solutions of this system are called theintegral curvesorintegral trajecto-
riesof the dynamical system. Therefore, an integral trajectory is a curveγ(t)
onMwhose velocity vector ̇γ(t) coincides withξ(γ(t)) for allt.
Theorem 3.A closed connected smooth manifoldMadmits a non-vanishing
tangent vector fieldξif and only if its Euler characteristic equals zero.
Corollary 1.
(a) The torusT^2 is the only orientable surface admitting a non-vanishing
tangent vector field.
(b) The Klein bottleK^2 is the only non-orientable surface with a non-vanishing
vector field.
Acknowledgements
I cordially thank Natalia Dobrinskaya, Taras Panov, and my wife Galina for
their great help in preparing these notes for publication.
References
- Victor M. Buchstaber, Taras E. Panov,Torus Actions and Their Applications
in Topology and Combinatorics, University Lecture, vol. 24 (American Mathe-
matical Society, Providence, RI, 2002) - Sergei P. Novikov,Topology I, Encyclopaedia Mathematical Science, vol. 12
(Springer, Berlin Heidelberg New York, 1996)