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Jul
2
awarded  Curious
Jun
16
answered Examples of “Unusual” Classifications
May
8
comment Deformation of Hitchin-Simpson correspondence
3. A holomorphic quadratic differential $\alpha$ determines in a natural way a holomorphic Higgs field $\Psi_\alpha\colon L\oplus L^*\to KL\oplus KL^*$ whenever $L^2=K.$ Then you add this Higgs field onto the Higgs field $\theta.$
May
8
comment Deformation of Hitchin-Simpson correspondence
2. You should try to compute what happens to first order in $t$ when you look at the solution to the self-duality equations corresponding to $t\alpha.$ This gives you a section $\bar\alpha\in\Gamma(\Sigma,\bar K K^{-1})$ (w.r.t. the hyperbolic metric) which can be considered as the tangent vector given by the variation of Riemann surface structures.
May
8
comment Deformation of Hitchin-Simpson correspondence
1.in every conformal class of metrics on a cpt. or. surface of genus $g\geq2$, there is a unique metric of constant curvature -4. Moreover, every metric which is not compatible with a given complex structure on your smooth surface gives a different Riemann surface structure on the surface.
May
7
answered Deformation of Hitchin-Simpson correspondence
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 $f_*A.$