For questions about surfaces.

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the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
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1answer
92 views

Homology subgroups generated by non-intersecting cycles

Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by ...
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1answer
35 views

Surface described by parametric equations

If I've got the surface in $\mathbb{R}^3$ described by: $x(s,t)=s^2-t^2$, $y(s,t)=s+t$, $z(s,t)=s^2+3t$ for $(s,t)\in\mathbb{R}^2$, and I'm told this surface is the graph of a function $f(x,y)$, how ...
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13 views

Scalar potential, vector field and line integral

I've been trying to get my head around this topic for a project where I need to reconstruct a 3D surface given the estimated normals of it (Photometric Stereo). I just want to be sure I'm ...
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1answer
23 views

Normal lines of a regular surface

I need to prove this: If all the normal lines to a regular surface pass through a fixed point, then the surface is a portion of the sphere. I haven't really tried much since I don't know what to do. ...
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12 views

Show that $\alpha_j=2iA_j$

I have a question, I don't understand that Why do we have $\alpha_j=2iA_j,\ P_j(z, \overline{z})=\text{Re}A_jz^j$ since proof of Lemma??? ======================================================= ...
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1answer
34 views

Questions about surface integrals and an example problem

It is- Double integral(x + y) dS Where S is the part of the cylinder y^2 + z^2 = 4 . With x being between 0 and 5 First question, if we want to get the integral of the surface of a cylinder, I ...
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1answer
35 views

oriented surface of genus g and m punctures

Let $S_{g,m}$ be an oriented surface of genus g and m punctures, what's the condition to ensure $S_{g,m}$ is hyperbolic? If $g\ge 2$, I know it is hyperbolic, how about g=0 and g=1? Thanks in advance. ...
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1answer
11 views

Double integrals using cylindrical coordinates

Suppose we had a vector field $F = (axy^2,ayx^2,x^2\cos(\pi z))$ and wanted to calculate the surface integral of $$\int\int_SF\cdot n \ dS$$ where $S = \{(x,y,z): x^2 +y^2 = 1, 0 \leq z \leq 1/2 \}$ ...
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23 views

Parametric equations of surfaces of revolution

I need to find a parametrization for the following two surfaces: (i) The surface obtained by revolving the curve $z = f(x),\ (a \lt x \lt b)$ in the $xz$ plane, around the $z$ axis, where $a \gt 0$. ...
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21 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
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1answer
37 views

Whats the general idea for finding the mapping class group of the twice punctured sphere?

Consider the twice punctured sphere. Now I know the mapping class group of the twice punctured sphere is $\mathbb{Z}_{2}$, the cyclic group of order 2. However, I know one applies the alexander trick ...
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2answers
121 views

How to parametrize a triangle?

How do I parametrize a triangle with vertices $A(1,1)$, $B(2,2)$ and $C(1,3)$? I have tried working with the equations of the lines that form it but am not completely sure how to link them together ...
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1answer
39 views

Metric Questions Using the First Fundamental Form

Consider the Poincare Disk with local coordinates $\rho=\log\frac{1+r}{1-r}$ and $\theta$, where $r$ is the radius and $\theta$ is the angle with $x$ axis. After finding the metric tensors as ...
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56 views

Showing that two surfaces are not isometric/locally isometric

I am trying to solve an exercise which asks to show that two surfaces are not isometric and additionally that they are not locally isometric. The two surfaces presented are graphs. I know that if two ...
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1answer
26 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
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101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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56 views

compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
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1answer
52 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
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2answers
130 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
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1answer
59 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
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1answer
21 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
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36 views

Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
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1answer
133 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
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1answer
26 views

Surface parameterization of a cylinder?

Suppose I wanted to parameterize the cylinder $x^2 + y^2 = R^2$ (for the purpose of computing a surface integral). Say $z$ is in range $-z_0 \le z \le z_0$. The standard parameterization I see ...
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15 views

How to calculate the double tangential derivative of a surface?

Let z=u(x,y) is smooth surface in $\mathbb{R}^3$. $\vec{\tau}$ is a tangential vector which is orthogonal to the vector of gradient direction $\nabla u$. How to prove the equality $$ u_{\tau\tau} = ...
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1answer
39 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
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1answer
42 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
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43 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
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2answers
68 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
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1answer
49 views

Find normal vector for the surface F(x,y)=0

I need to find the normal vector(in a point (a,b)) for a surface F(x,y)=0, that we can't write as y=f(x) and F(x,y) doesn't satisfies the conditions of the implicit function theorem. For example: the ...
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1answer
53 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
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1answer
101 views

Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
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1answer
41 views

Orthogonal Parametrization of a Regular Surface

I was just wondering whether or not it is always possible to parametrize a regular surface $S$ via a function $X$ of local coordinates $u$, $v$ such that $X$ is an orthogonal parametrization- that is ...
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1answer
32 views

GNU Octave draw figure of 2 planes

How can I draw two planes in same figure in GNU Octave? $$ x + y + z = 1\\ 2x - y + 3z = 4$$ Thanks!
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31 views

Show that the set is a surface

I would be appreciate some help with the following problem, which has been suggested (but not assigned), by my professor: Consider the mapping $f : \mathbb R^3 \rightarrow \mathbb R$ defined by ...
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1answer
43 views

Surfaces of Revolution with Constant Gaussian Curvature

Surface $S$ is parametrized by $$X(u,v) = (\varphi(v) \cos{(u)}, \varphi(v) \sin{(u)}, \psi{(v}))$$ with everywhere-constant Gaussian curvature $K$. Let $v$ be the arc length of the generating curve ...
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1answer
41 views

Proving the a Pseudosphere is a Regular Surface

I need to prove that a pseudosphere $S$ is a regular surface. I have found for $S$ the parametrization $X(t,\theta) = (\mathbb{e}^{t} cos{(\theta)}, \mathbb{e}^{t} sin{(\theta)}, \pm ...
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1answer
40 views

Proving that a surface is a Möbius strip

I have a given parametrization $X(u,v)$ of a surface $S$ in $\mathbb{R}$. I must prove that it is a Möbius strip. I cannot use graphical means and I am not to reparametrize the surface- essentially, I ...
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Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
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29 views

How does this integration by parts work: $\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$

Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$. The normal vector to $S$ at ...
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52 views

Surface Area cut from cone by cylinder

I genuinely have no idea how to do this. I've been struggling on this problem for about two hours now (more hours last night), and am not getting anywhere. The question is: Find the surface area ...
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1answer
53 views

Does the connected sum depend on direction of gluing?

The connected sum of two surfaces (2-manifolds) is defined by removing a disk from each and gluing the cut edges: (Image adapted from Wikipedia) Does the resultant surface (up to homeomorphism) ...
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1answer
77 views

Topological surface thought experiment

Imagine a two-dimensional version of you lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you ...
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1answer
58 views

The principal curvatures of a surface of revolution

The principal curvatures of the surface at a point is defined as the maximal and the minimal curvature among all normal sections. It's claimed (say, on Stillwell's Geometry of Surfaces) that for a ...
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2answers
130 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
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1answer
34 views

Why use Gauss and mean curvature to characterize a surface's deviation from being “flat” at one point?

We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum ...
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Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
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1answer
102 views

Showing to be Unit-sphere

First and second fundamental forms are both $du^2 +\cos^2 u dv^2$ I want to show that the surface is a part of the unit sphere. What I did is following; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
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33 views

Example of 1-dimensional hypersurface in $\mathbb{R}^2$ which is compact?

Is there an explicit example of a $1$-dimensional $C^k$ hypersurface in $\mathbb{R}^2$ which has no boundary and is compact? I know of a circle, but want something like an interval.