Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.