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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 multi-linear 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 CHern-Simons 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 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