5
votes
0answers
76 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
1
vote
0answers
53 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
5
votes
0answers
116 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
3
votes
1answer
90 views

projective cubic curve to complex projectie space

Suppose we are given the equation $$ y^2z = x(x - z)(x - 2z) $$ I would like to define a degree two map $g$ on this curve into complex projective space. I hate to say I am already lost here - how do I ...
2
votes
0answers
57 views

Torus biholomorphic to smooth cubic curve?

I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ ) I think I ...
6
votes
2answers
483 views

$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
2
votes
0answers
95 views

Partition of Unity for the Divisor Sheaf

Recall that given a Riemann Surface $X$, the divisor sheaf is the sheaf ${\cal D}$ which assigns to each open set $U$ the collection of maps $\phi:U \to \mathbb{Z}$ such that $\phi(p)=0$ for all but ...
6
votes
1answer
280 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, ...
0
votes
1answer
250 views

Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
4
votes
2answers
367 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 ...
0
votes
1answer
122 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 ...
4
votes
1answer
320 views

Local normalization of algebraic curves

I am currently reading about the normalization theorem: Suppose $C$ is an irreducible plane algebraic curve, and let S be the set of singular points. Then there exists a compact Riemann surface $\hat ...