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comment Darboux-like coordinates on a Kähler manifold
Generally speaking, it's the complex structure that's most important to keep track of easily. So the most common coordinates used in calculations are holomorphic coordinates $z^1, \dots, z^n, \bar{z}^1, \dots, \bar{z}^n$, and the Kähler metric is represented using its potential function. Normal coordinates are more useful in real Riemannian geometry, because they are connected to Jacobi fields. I can imagine that in some circumstances Darboux coordinates are useful. Another alternative is to use a unitary frame of $(1,0)$-forms and not coordinates. Try all of them and see which works best.
Feb
5
comment No normal coordinates on general Finsler manifolds
So there are $C^1$ normal coordinates?
Feb
5
comment No normal coordinates on general Finsler manifolds
Could you provide what the definition of "normal coordinates" would be for a Finsler manifold?
Feb
5
comment Is the Flajolet-Martin constant irrational? Is it transcendental?
Jeff is an old-fashioned guy. It also follows the tradition of Donald Knuth.
Feb
2
comment Please help me with my assignment. Thank you
This site is not for homework. Please consult your teacher or teaching assistant.
Jan
31
comment Is there a Nash-type theorem for symplectic manifolds?
Isn't $\omega$ being exact a necessary condition for the existence of an embedding?
Jan
31
comment geodesic computation: “energy” minimization versus arc length minimization
That sounds reasonable, but what does that have to do with the energy functional? It is indeed straightforward to write down the ODE satisfied by a geodesic parameterized by one of the coordinates, and indeed the ODE is not the Euler-Lagrange equation of the energy functional. That's not surprising since the energy functional is not invariant under changes of parameterization (in contrast to the length functional).
Jan
30
comment geodesic computation: “energy” minimization versus arc length minimization
Why would you want to restrict the parameter like that?
Jan
30
comment geodesic computation: “energy” minimization versus arc length minimization
You should move this question to math.stackexchange.com. A parameterized curve in a Riemannian manifold that is a critical point of the energy functional is always a geodesic parameterized by a constant times arclength.
Jan
26
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Jan
26
comment Canonical Metric on Grassmann Manifold
To repeat what Sebastian already said, the question refers to Grassmannians, which most of us would interpret as real Grassmannians, i.e., $G(n,k) = $ the space of $k$-linear subspaces in an $n$-dimensional real vector space. In general there is no complex structure on such a space, for example if the Grassmannian is odd-dimensional. However, it is a homogeneous space, and the standard meaning of a "canonical metric" on a homogeneous space is one that is invariant under the group action.
Jan
25
comment Obtaining Hessian of the embedding from an induced metric
A simple example is $z = x^2$. The induced metric is the standard flat one, also induced by $z=0$, but the two embeddings have different Hessians.
Jan
22
comment Does a Riemannian manifold have a triangulation with quantitative bounds?
If you can read German, the original paper is: here: link.springer.com/article/10.1007%2FBF01168235
Jan
22
comment Does a Riemannian manifold have a triangulation with quantitative bounds?
Could the almost-linear co-ordinates introduced by Jost and Karcher provide a means to proving this? You can find an English explanation of these co-ordinates in the following lecture notes: Jost, Jürgen Harmonic mappings between Riemannian manifolds. Proceedings of the Centre for Mathematical Analysis, Australian National University, 4. (I might be able to find my copy of it, if you think this would be useful)
Jan
20
comment Upper bound for Willmore energy
It suffices to compute or estimate the Willmore energy for an explicit embedding of the surface. I imagine there are a number of ways to do this. One possible approach for a surface of genus $g$ is to embed $g$ copies of the torus explicitly and glue them together by removing small disks and attaching a small cylinder that flattens out at both ends. The total Willmore functional of the surface is roughly equal to the sum of the Willmore functionals of the $g$ tori and $g-1$ cylinders. These can be estimated using the explicit formulas defining them.
Jan
16
comment An example for affine function
What does "affine function" mean in this context?
Jan
13
comment Conformally flat manifold with zero scalar
"Conformally flat" = "locally conformally flat".
Jan
13
comment Conformally flat manifold with zero scalar
In dimensions 4 or higher, conformally flat means the Weyl tensor vanishes. If the scalar curvature also vanishes, there is still the possibility that the trace-free part of the Ricci curvature is non-vanishing. As Robert points out, there are indeed local metrics satisfying this.
Jan
11
comment Concentration compactness on a compact setting
The usual case is when the sequence is a minimizing sequence for an energy associated with a critical Sobolev inequality. One then analyzes the limiting situation via a blow-up argument (rescaling the manifold near a point where the sequence is converging weakly to a point measure).
Jan
11
comment Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
Thomas's remark, along with the definitions of the two spaces you're trying to decide between, immediately gives the answer. This is more appropriate for math.stackexchange.com