How to construct a vector fields with isolated zeros?

The Poincare-Hopf theorem tell us that the sum of the indices of a vector field at isolated zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold. But how to construct a vector fiedls with isolated zeros?

Your question isn't very well defined. A manifold on its own is not an object where constructions come by easily. But there is a generic way to construct vector fields with isolated zeros. Any vector field can be approximated by one with isolated zeros. This is a consequence of Sard's theorem. So start off with the zero vector field and choose any small random perturbation of that, and there you go.

If you want a more constructive answer you'll have to assume a more constructive situation. Like say if your manifold is triangulated, or has a handle decomposition, or a morse function.

Chapman describes the Morse situation so I'll give the triangulation situation.

The vector field has these properties:

There is a critical point at the barycentre of every cell in the triangulation. The vertices are repellors. The barycentres of the top-dimensional simplices are the attractors. A 1-simplex is a (1,n-1)-index critical point -- meaning there's two orbits approaching (along the 1-simplex) and an n-2-dimensional family of reverse orbits attracting. Etc. A j-simplex barycentre has a j-1-dimensional family of attracting orbits, and an n-j-1-dimensional family of reverse orbits attracting.

That isn't quite explicit as one needs an explicit smoothing of the triangulation to put this all together. But it gives you the idea.

• I remember first reading about this construction in Hopf's Differential Geometry in the Large. It's a great book for concrete insights like this one. – Per Vognsen Jul 28 '10 at 7:16

If one takes the differential of a Morse function, one gets a differential form (a cotangent field) with isolated zeros. If one has a Riemannian metric on the manifold one can convert between covector fields and vector fields. So, from a Riemannian metric and a Morse function you can write down a vector field with isolated zeros.

Just use the transversally theorem, an application of Sard's theorem: the generic vector field intersects the zero-section of the tangent bundle transverse, therefore the zeros are isolated.