# Tagged Questions

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### The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
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### Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface. (You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
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### $\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
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### Two questions on isometric embedding

According to the answer of the following question, I try a new version: An special isometric embedding Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic. Is there an ...
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### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result: The geodesic flow on a manifold with negative curvature is ergodic. The lecture note that ...
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### Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic n-space. Let $TH$ be the tangent bundle of $H$, endowed with its Sasaki metric. I have two questions: Is $TH$ isometric to $H$ times a flat n-space? What is the group of ...
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### Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
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### Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question: "Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?" I believe the answer to ...
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### Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?

First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...
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### The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations, Theorem 6.1 (Selberg Trace formula) on page 26 of these notes. Equation 3.19 and 3.20 on page 11 of this paper. I vaguely feel that these two are the ...
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### Negative sectionnal curvature and constant curvature

Good morning everyone, I was wondering about the difference between manifolds carying a Riemanniann metric with negative sectionnal curvature and hyperbolic manifolds. I was told once "there are ...
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### Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat. How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy: $$-\Lambda \le K \le -\lambda$$ for ...
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### Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this case the universal cover is the hyperbolic plane ...
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### Volume of a Riemannian manifold and its relation to fundamental group

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
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### Tweetable way to see that Willmore energy is Möbius invariant?

Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional $$\mathcal{W} = \int_M H^2 dA$$ ...
In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ... 3answers 637 views ### F→E→B bundle with B,E,F hyperbolic: possible? It would be interesting to me obtain an answer to the following easy to state question: Does there exist a (smooth) fibre bundle$\pi\colon E\rightarrow B$with typical fibre$F$such that$E$,$B$... 1answer 192 views ### altering curvature on a tessellation representation of a compact surface I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ... 1answer 1k views ### Pythagorean Theorem for Right-Corner Hyperbolic Simplices? My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra: Euclidean:$A^2+B^2+C^2=D^2$Hyperbolic: ... 2answers 594 views ### Reference for the geometry of horospheres Dear all, I am looking for a reference to a proof of the following well-know fact (cited for example by B.Farb in Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840). ... 2answers 469 views ### Schwarz Lemma in terms of conformal surfaces or holomorphic curves? Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting. Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ... 2answers 2k views ### A question about the proof of Mostow rigidity I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism$\alpha: \Gamma ...
Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...