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Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

12 votes
3 answers
985 views

Area of a smooth complex projective curve

Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\math …
3 votes
0 answers
119 views

Which holomorphic curves can be leaves of a non-singular holomorphic foliation of $\mathbb C...

It is easy to see that for any entire function $f : \mathbb C \to \mathbb C$, its graph $G(f) = \{(z,f(z)) \in \mathbb C^2 \mid z \in \mathbb C\}$ can be translated by $(0,c)$ for any $c \in \mathbb C …
2 votes
1 answer
153 views

Curvature of curves through a point of a surface smoothly embedded in Euclidean space

The curve C(𝜃) drawn on a smoothly embedded surface 𝜮 in 3-space — where C(𝜃) is defined as the intersection of 𝜮 with a 2-plane perpendicular to 𝜮 at P — leaving the point P at angle 𝜃 will hav …
5 votes
2 answers
396 views

Gaussian curvature of a holomorphic curve in complex 2-space

Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$. Each point of $M$ has …
5 votes
3 answers
343 views

Classification of surface bundles over surfaces

Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces? Now, if the fibre F and the base B are both Hausdor …
3 votes
1 answer
181 views

Identifying the conformal equivalence class of a 2-torus subgroup of the cubical 3-torus

Let K, L, M be integers with gcd(K,L,M) = 1. They determine a connected Lie subgroup G = G(K,L,M) of the cubical 3-torus (ℝ/ℤ)3 via G = {(x,y,z) ∊ (ℝ/ℤ)3 | Kx + Ly + Mz = 0} (where 0 denotes the i …