Sam Lisi
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 Mar 27 comment Winding number of vector field on surface It's not more general... what I described above really uses the fact that the structure group of the tangent bundle to an oriented surface can be reduced to U(1). This is not true for a general manifold. In some sense, what I describe can be generalized to a discussion of Maslov classes of Lagrangians in symplectic manifolds... but that's much fancier than what you are interested in. Mar 26 answered Winding number of vector field on surface Mar 26 comment Winding number of vector field on surface Do you know anything about the zeros of these vector fields? These can make the discussion a little tricky. Mar 22 comment Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$ My hint would be to think about what a derivation has to be, if you look at it in a coordinate chart. Another hint is to remember that you have cut-off functions and partitions of unity. Mar 22 comment Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$ The existence of $\chi$ is key in differential geometry. See for instance en.wikipedia.org/wiki/Bump_function. Mar 20 comment How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$? Another idea would be to take a Morse function with a unique minimum. Then, take the union of all the ascending manifolds from higher index critical points. This is of codimension 1 and thus of measure 0. I'm pretty sure the complement is the ascending manifold of the minimum and thus diffeomorphic to a ball. Oh, wait, that's the link that Jason gave. Sorry. Mar 20 revised what are the holomorphic curves in $T^{*}S^3$ with boundary on the zero section? added 1407 characters in body Mar 20 comment what are the holomorphic curves in $T^{*}S^3$ with boundary on the zero section? I recommend a nice survey article (though a little obsolete) by Hofer and Kriener. I'll update what I wrote with a few more details. Mar 20 comment Vector field and integral curve You know the total expression needs to be independent of the extension of $v$. However, you can extend $v$ however you want... what if you take an extension of $v$ that is invariant under the flow of $V$? Once you've seen what happens in that case, you can try to understand what happens more generally, though that case is sufficient to prove what you want. Mar 20 answered what are the holomorphic curves in $T^{*}S^3$ with boundary on the zero section? Mar 20 comment Vector field and integral curve After you've extended the vector field, you want to expand the LHS as $= \nabla_V(d\phi_t \cdot v)|_{t=0} = \nabla_{d\phi_t \cdot v} V|_{t=0} + [V, d\phi_t \cdot v]|_{t=0}$ first using the definition of covariant derivative and then using the fact it is torsion-free. You now want to rewrite this last term as $L_V(d\phi_t v)$ and evaluate it. Mar 19 comment Vector field and integral curve Indeed, I was thinking of $v$ as being a local vector field. You can always extend your vector $v$ to a vector field and then show the result doesn't depend on your extension. (So, no, you can't prove $\nabla_V v = 0$, since that will depend on how you extend $v$.) Mar 18 answered Vector field and integral curve Mar 16 awarded Yearling Sep 17 awarded Citizen Patrol Sep 17 comment When is a $k$-form a $(p, q)$-form? @MichaelAlbanese: that's how it always is with these things... impossible before you see it the right way and obvious afterwards. Sep 16 answered Integrals on manifolds and pullbacks Sep 16 comment Construction of cut-off function Can you please verify what you mean instead of (3)? As I commented, it is impossible to satisfy it on the region where $2r < x < r$. The tricky part of proving that a cut-off function with these properties exists comes from the differential inequalities (3) and (4). One way to start is to see what functions are extreme cases of (3) and (4). This gives an idea of the type of behaviour the function must have and where the constraints have a chance to be problematic. Have you tried this? Let us know what you have tried to do. Sep 16 answered When is a $k$-form a $(p, q)$-form? Sep 16 comment Construction of cut-off function Why are you interested in this question? Do you want an argument that such a function exists, or do you want an explicit formula? Also, is there a typo in condition (3) for the interval $-2r < x < -r$? On that interval, there must be points where $\phi'(x) > 0$.