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2h

comment 
Calculating Exterior Distance from Measurements of Inner Geometry
I guess that is why Thomas wrote in his answer "a piece of a cylinder", and why I explicitly specified my subsets of the plane and of the cylinder in my answer below. 
3h

answered  Calculating Exterior Distance from Measurements of Inner Geometry 
1d

comment 
Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic?
@Michael: You are of course right, that one can recover the multilinear algebra. Nevertheless, this is not totally transparent, and quite often people prefer to use connections in order to work with higher order derivatives. 
Oct 17 
answered  Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic? 
Oct 14 
comment 
An example of mean curvature flow that does not preserve embeddedness
Have you tried a trefoil knot? I am also pretty sure that you can connected your 2 circles "far out" such that you can get an example of the mean curvature flow which does not stay embedded (before singularities occur). 
Oct 7 
comment 
Symplectic form on moduli space of connections
I do not know an answer for the general linear case, but for compact groups like $SU(2)$ there is a nice answer, see for example "Some comments on CHernSimons Gauge Theory" by Ramada, Singer and Weitsman. 
Sep 26 
answered  Factors of automorphy from Chern connection 
Sep 23 
revised 
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
added 429 characters in body 
Jul 2 
awarded  Curious 
Jun 16 
answered  Examples of “Unusual” Classifications 
May 8 
comment 
Deformation of HitchinSimpson 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 HitchinSimpson correspondence
2. You should try to compute what happens to first order in $t$ when you look at the solution to the selfduality 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 HitchinSimpson 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 HitchinSimpson 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 2sphere
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 