a friend asked me to pose the following problem:
It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler characteristic class is non-zero. I want to construct such a vector field.
In his book, Comprehensive Introduction to Differential Geometry vol I, Mr. Spivak suggests the following steps:
- We take a triangulation of the manifold.
- In the interior of each n-simplex we construct a vector field with only one zero to obtain a vector field over the whole manifold with an infinite, but discrete, discret zeros (precautions to take in interfaces?)
- We join all the zeros of this field with a path.
- Take a tubular neighborhood of this path.
- Finally, using an isotopy to "hunting" all zeros to infinity to obtain the desired field.
I tried this idea, writing a detailed demonstration without success. Could you help me to do?
One could also try to build on the manifold a function without critical point, then take a Riemannian metric and using the gradient. I do not do well. Is there a demonstration of such a fact?
Thank you for any suggestions.