Sam Lisi
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 May7 awarded Revival May7 awarded Caucus May7 answered When does a vector field admit orthogonal fields? May7 answered An alternative description of the first Stiefel-Whitney class May4 comment When does a vector field admit orthogonal fields? @Yvoz How does this last observation of yours not answer your question? It seems like a characterization of when such a vector field exists. What are you looking for if that isn't the answer? May3 comment When does a vector field admit orthogonal fields? Observe that you don't need orthogonality for the second vector field -- you just need pointwise linearly independent and nowhere vanishing. Then, since $\mathbb{R}^3$ is oriented, you get a third vector field free. You are therefore trying to find a trivialization of the tangent bundle of the quotient of $U$ by the flow of $X$. In general, this quotient will be a horrible space and won't have a tangent bundle, but maybe something can be said anyway. (Note that in your two examples, the quotients are $\mathbb{R}^2$ are $S^2$ respectively, and in particular very nice.) May3 comment An alternative description of the first Stiefel-Whitney class I was not thinking about Steenrod squares etc. I really was thinking about their chapter on obstruction theory, Chapter 12. I guess M&S refer back to Steenrod's book, which I haven't studied as carefully as I should have. On page 143, they explain how you can reconstruct the obstruction classes from the $w_i$, at least for small $i$. May3 comment An alternative description of the first Stiefel-Whitney class @Mehdi : yes, that's what I was thinking about (to your first comment). I'll try to write something soon, but the short version of what I am thinking about is that the $\mathbb{Z}/2$ that shows up in the sequence $1 \to SO(n) \to O(n) \to \mathbb{Z}/2 \to 1$. May3 comment An alternative description of the first Stiefel-Whitney class What definition(s) of $w_1$ do you know? If you have been studying Milnor-Stasheff, take a look at the obstruction theory chapter. May1 answered Question about symplectic tranformations May1 comment An other question about Theorem 3.1 from Morse theory by Milnor May1 comment An other question about Theorem 3.1 from Morse theory by Milnor Do you understand what $\phi_t$ is? Apr30 comment An other question about Theorem 3.1 from Morse theory by Milnor The tricky part is that $\phi_{b-a}$ maps $M^a$ to $M^b$. To check this, calculate what $f( \phi_{b-a}(x))$ is, if $f(x) \le a$. This is where you use the fact that your new vector field flows downwards so that the time $t$ flow changes the value of $f$ by $t$. Apr30 comment An other question about Theorem 3.1 from Morse theory by Milnor The fact that it is a diffeomorphism onto its image follows just from the existence/uniqueness of ODE, the long-time existence (essentially by construction) and smoothness with respect to initial condition. I.e. this is standard ODE theory. Apr30 comment An other question about Theorem 3.1 from Morse theory by Milnor The purpose of (1) is to make (2) obvious and to make it easy to write down a formula for $r_t$. Can you construct a simple example (e.g. for domains in $\mathbb{R}^2$) and see what he is doing? Can you say more about where you are stuck? Apr30 answered Implicit Function Theorem and Rank Theorem Misunderstandings. Apr30 comment Dynamical Systems. Bendixson's and Dulac-Bendixson's theorems. Great timing, I was just typing essentially the same thing. While this is clear from what you wrote, I want to emphasise that $U$ has positive area because $C$ is embedded. ($C$ is embedded by uniqueness of solutions to ODE.) Apr27 answered Non-degenerate solutions to constant Hamiltonian flow Apr27 answered Lagrangian subspaces Apr16 comment Area of flux homomorphism in symplectic topology I don't understand what you are trying to do, specifically. Are you trying to compute this explicitly? for that to make sense, it seems to me that you need some more explicit information. Or are you trying to do something else? this isn't clear to me.