# Tagged Questions

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### Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
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### Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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### equivariant map $\mathbb{CP}^1 \to \mathbb{CP}^2:$ where does it send the boundary

Start from a map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$x = au^2, \quad y = av^2, \quad z = uv,$$ where $a \in \mathbb{R}$ is a parameter. The image of the map is the ...
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### The Definition of the Second Fundamental Form

Let $r:M\rightarrow{\mathbb{R}^{n+1}}$ be an isometric immersion and $M$ is an $n$-dimensional Riemannian Manifold. That is to say, $M$ is the hypersurface in $\mathbb{{R}^{n+1}}$. Then we can ...
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### Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
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### Showing that a map between surfaces is a local isometry

Currently studying Differential Geometry of Curves and Surfaces. We have: $$\sigma:(0,\infty) × \mathbb{ R} \to \mathbb{R}^3, \quad (u,v) \to \frac{1}{\sqrt{2}} (u \cos v,u\sin v ,u)$$ We need to ...
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### Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
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### Show that there is a ﬁxed $p \in \mathbb{R}^n$ such that for all $s \in I, \gamma(s)=\beta(s)+p$.

Suppose that $\beta,\gamma : I \to \mathbb R^3$ are two unit speed smooth curves. Suppose that the curvatures and tortions are everywhere positive, and that $B_\beta(s)= B_\gamma(s)$ for all $s\in I$. ...
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### Difference between tangent space and tangent plane

I’ve avoided doing any manifold (regretting it somewhat) courses, however do have some understanding. Let $p$ be a point on a surface $S:U\to \Bbb{R}^3$, we define: The tangent space to $S$ at $p$, ...
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### Deriving equations for the “Bianchi-Pinkall torus”

I am trying to work out explicit parametric equations for the "Bianchi-Pinkall flat torus" as depicted in this note, but I seem to have gotten stuck in understanding the descriptions given in that ...
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### How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
Let $S\subseteq\mathbb{R}^3$ a closed surface and let $X\in\mathfrak{X} (S)$ a vector field on $S$ such that $\mid\mid X_p\mid\mid \le M$ $\forall p\in S$ for some constant $M>0$. Prove that $X$ ...