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Mar
18
answered In what condition is a conformal flat manifold flat?
Mar
12
awarded  Yearling
Feb
28
comment quantitative version of the rigidity of the 2-sphere
I am not an expert on the Ricci flow, but can't you use the existence of the flow for positive curvature metrics and corresponding estimates to deduce "how long" you need to flow, i.e., how close you are to the round sphere?
Feb
27
answered A question on differential forms and integral invariants
Feb
7
awarded  Scholar
Feb
7
accepted Riemann's theorem on theta
Jan
27
comment Normalizing the value of a principal connection at a point
You both are right, I was not clear enough with my phrasing "connection 1-form with respect to this section vanishes at p," which should mean pullback of the connection form by the section. And Robert's remark shows us how nicely differential calculus can be approximated linearly, I just oversaw it.
Jan
27
answered Normalizing the value of a principal connection at a point
Dec
13
answered How to compute the normals to Costa's minimal surface?
Oct
29
comment Obstruction to this gauge choice of the connection of a vector bundle
Sorry, my mistake. Of course, you are right, as I should have been aware of if I had read the whole question.
Oct
28
comment Obstruction to this gauge choice of the connection of a vector bundle
Somehow, a short computation shows me that your equation $\mathcal L_RA=d_A(I_R A)$ is not gauge invariant, and therefore not even well-defined on an arbitrary vector bundle
Sep
22
awarded  dg.differential-geometry
Aug
1
awarded  Nice Answer
Jul
25
comment Formula for the curvature of an induced connection
If $a=0$ then $F_B=\lambda\cdot F_A$ where $\lambda\cdot\colon\mathfrak g\to \mathfrak h$ is the induced map on Lie algebras. For $a\neq0$ you would in general need also to know $f_*A.$
Jul
22
comment $E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
Over Riemann surfaces there is a proof of this fact which does not use Newlander-Nirenberg: Atiyah, Bott: The Yang-Mills equations over Riemann surfaces.
Jul
22
comment $E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
Dear Tim, I think it is the same complexity to show that flat connections are locally trivial and that flat Riemannian manifolds are locally euclidean: Take a ON parallel frame of the Riemannian manifold, because the connection is torsion free these vectorfields are commuting, so you can integrate them up to obtain a local isometry to euclidean space.
Jul
11
asked Periods of translation surfaces
Jun
25
awarded  Revival
Mar
12
awarded  Yearling
Mar
5
comment Branched Regular Cover over 4-times punctured sphere
Sorry, I was to vague about this: there exists holomorphic coordinates $y$ and $z$ on $\Sigma$ and $P^1$ centered at a branch point resp. branch image such that $f$ is given by $z=y^{g+1}.$ Then, the 4 sheets are exchanged by going around $y=0,$ and such a deck transformations $\varphi_{g+1}$ extends to all of $\Sigma$ holomorphicallysuch that $f\circ\varphi_{g+1}=f.$ This topic is covered in most Riemann surface books, for example the recent one by Donaldson.