How to construct a vector fields with isolated zeros? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:30:21Z http://mathoverflow.net/feeds/question/33622 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33622/how-to-construct-a-vector-fields-with-isolated-zeros How to construct a vector fields with isolated zeros? Chen 2010-07-28T06:33:51Z 2010-07-28T06:45:44Z <p>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? </p> http://mathoverflow.net/questions/33622/how-to-construct-a-vector-fields-with-isolated-zeros/33623#33623 Answer by Robin Chapman for How to construct a vector fields with isolated zeros? Robin Chapman 2010-07-28T06:40:30Z 2010-07-28T06:40:30Z <p>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.</p> http://mathoverflow.net/questions/33622/how-to-construct-a-vector-fields-with-isolated-zeros/33624#33624 Answer by Sebastian for How to construct a vector fields with isolated zeros? Sebastian 2010-07-28T06:44:29Z 2010-07-28T06:44:29Z <p>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.</p> http://mathoverflow.net/questions/33622/how-to-construct-a-vector-fields-with-isolated-zeros/33627#33627 Answer by Ryan Budney for How to construct a vector fields with isolated zeros? Ryan Budney 2010-07-28T06:45:44Z 2010-07-28T06:45:44Z <p>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. <em>Any</em> 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. </p> <p>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. </p> <p>Chapman describes the Morse situation so I'll give the triangulation situation. </p> <p>The vector field has these properties: </p> <p>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. </p> <p>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. </p>