Questions tagged [conformal-geometry]
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10 questions from the last 365 days
2
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125
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How does a conformal transformation affect the frame bundle metric of that manifold?
Suppose I have a metric $g_{\mu\nu}$ over an n-dimensional smooth orientable Riemannian manifold $M$. We then utilize Cartans repere mobile (moving frames) to define oriented orthonormal frames $e^{a}=...
- 250
2
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1
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196
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A question in Möbius geometry [closed]
I am currently studying Möbius geometry from the book [1]. I found a group in Möbius geometry called Möbius group which contains Möbius transformations. I have the following doubt.
Dose this group ...
- 121
0
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0
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71
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Geodesic distance under conformal perturbation
Let $(M,g)$ be a complete Riemannian manifold of dimension $d\ge 3$. Suppose that $g_0$ is another Riemannian metric on $M$ which is conformal to $g$; i.e. $g = e^{2u}g_0$ for some $u\in C^{\infty}(M)...
- 5,405
2
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0
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56
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Convergence of conformal metrics with prescribed curvature
We know that for any function $K: \mathbb{D} \to \left[-a, -b\right]$, where $a, b > 0$, there is a unique metric $h$ on the disk $\mathbb{D}$ which is conformal to $dz^{2}$, and has curvature ...
- 63
2
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0
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57
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A sequence of conformal metrics with bounded negative curvatures on the disc
Let $\mathbb{D}$ denote the unit disk, and let $h_{-1}$ be the unique hyperbolic metric on $\mathbb{D}$ which is conformal to $dz^{2}$.
Take a sequence of smooth complete metrics $h_{n} = e^{\rho_{n}} ...
- 63
1
vote
1
answer
361
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Can the Causal Structure recover the manifold topology for non-chronological spacetimes?
Given a time-oriented spacetime $(M,g)$, a binary relation $\ll$ can be defined on this spacetime where $p \ll q$ for $p, q \in M$ if and only if there exists a time-like path connecting $p$ and $q$.
...
- 612
2
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0
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47
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Growth/Decay of conformal Killing fields in cone metrics
Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric
$$g= dr^2 + r^2 \gamma$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing ...
- 969
2
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1
answer
103
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Parameterizing Teichmüller spaces of punctured surfaces
Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
- 343
2
votes
1
answer
169
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A formula for the cross-ratio in terms of hyperbolic data
Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.
We will use the following convention for the cross-ratio $CR$ of ...
- 5,215
2
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0
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104
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Convergence of diffeomorphisms
Let $(\Sigma, g)$ be a compact $n$-dimensional Riemannian manifold without boundary. Let $F_i$ be a sequence of diffeomorphisms of $\Sigma$ and $u_i$ be a sequence of positive scalar functions.
...
- 169