Tagged Questions

1
vote
1answer
36 views

Using Gauss-Bonnet to prove that geodesics have at most one point of intersection

Given an oriented Riemannian manifold $(M,g)$ of dimension $2$, such that $M$ has negative Gaussian curvature everywhere and $M$ is diffeomorphic to $\mathbb R^2$, I'm looking for a way to show that ...
5
votes
2answers
87 views

Hopf's theorem on CMC surfaces

I got stuck reading the proof of the following theorem: Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere. Proof: Let ...
0
votes
0answers
30 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
3
votes
2answers
54 views

Invariance of curvature under a conformal mapping

Let $\Omega_{1}, \Omega_{2} \subseteq \mathbb{C}$ be bounded domains. Let $\rho$ be a metric on $\Omega_2$ and $h: \Omega_1 \rightarrow \Omega_2$ a conformal mapping. Let $$h^*\rho(z) = ...
5
votes
1answer
223 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
5
votes
1answer
120 views

About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
4
votes
2answers
143 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
5
votes
1answer
145 views

Parabolic elements correspond to punctures

In Mapping Class Group by Farb and Margalit page 22, they say: Let $S$ be a hyperbolic surface. If a non-trivial element of $\pi_1(S)$ is represented by a loop (up to homotopy) around a puncture, ...
0
votes
1answer
61 views

Defintion of totally geodesic flat submanifold

I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian ...
0
votes
1answer
232 views

The conditions of a metric to be geodesically complete

On $\{\vec{x}\in \mathbb{R^n}:x_1^2+x_2^2+\cdots+x_n^2<1\}$ in $\mathbb {R^n},$ what $\alpha$ can make the metric $$g=(1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes ...
2
votes
1answer
157 views

Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal ...
3
votes
2answers
86 views

How does one move a point in $B(0,1)$ to the origin with a Möbius transformation

Let $z_0$ be in the open unit disc $B(0,1)\subset \mathbf{C}$. Is there a general formula for an automorphism of $B(0,1)$ which sends $z_0$ to the origin? I find it easier to think about the complex ...
2
votes
0answers
201 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
1
vote
1answer
65 views

How does a (1,1)-form on a Riemann surface induce a metric on the space of holomorphic differential forms

Let $\mu$ be a (1,1)-form on a compact connected Riemann surface $X$ of genus $g$. (Assume $\mu$ to be a real positive smooth $(1,1)$-form if necessary.) The space $H^0(X,\Omega^1)$ of holomorphic ...
2
votes
1answer
219 views

Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature. For instance, What is the spectrum of the ...