For questions about surfaces.

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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.
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If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
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1answer
44 views

What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.
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1answer
14 views

If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$ Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)? I don't know what the chart map should be...
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50 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
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1answer
111 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
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58 views

How do you integrate surface area in spherical coordinates?

A single-valued function of spherical coordinates $r(\theta,\phi)$ (where $(\theta,\phi)\in[0,\pi]\otimes[0,2\pi]$) naturally defines a surface in 3D space. How does one calculate the surface area of ...
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2answers
86 views

How to parametrize this region surface

$S$ is the portion of the plane $$x+2y-3z=3$$ in the octan bounded by the positive direction of the $x$ and $y$ axis and the negative direction of the $z$ axis. How can I parametrize this crazy ...
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1answer
41 views

Can any 3d surface be mapped to a 2d plane?

I could see a 3D surface being parameterized by 2 parameters such that the surface is broken up into many infinitesimally thin curves. Is this true even for surfaces that might "fold" over itself or ...
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47 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
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1answer
182 views

Verify Divergence Theorem (using Spherical Coordinates)

I'm very used to calculating the flux of a vector field in cartesian coordinates, but I'm still getting tripped up when it comes to spherical or cylindrical coordinates. I was given the vector field: ...
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2answers
82 views

parametrize surface region

S is the elliptic region of the plane $y+z=1$ inside the cylinder $4x^2+4(y-0.5)^2=1$. First parametrize $S$ using $(x,y,z)=G(u,v)$ and then calculate $\displaystyle \frac{dG}{du}\times ...
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2answers
60 views

parametrize a disc

$S$ is the disc of radius 1 centered at the origin located on the $xy$ axis, oriented downward. First parametrize the given surface using $(x,y,z)= G(u,v)$ with $(u,v)$ in $W$ and then calculate ...
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1answer
33 views

Surface with border is homotopy equivalent to bouquet of circles

Why is any compact surface with non-trivial boundary homotopy equivalent to bouquet of circles? It was mentined in "Course homotopy topology" by Fomenko, Fuchs while calculating homotopy groups of ...
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1answer
120 views

a good introduction to Laplace Beltrami operator over differential manifolds?

I'd like to have a good reference to understand how the Laplacian operator get generalized over differential manifolds. More concretely, I want to understand and prove the equation : $$\Delta ...
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1answer
47 views

Trying to prove shortest distance between two points

I'm trying to prove that the shortest distance between two points in the Euclidean plane is a straight line: Here is what I've achieved so far; but I've got lost right at the end if anyone could ...
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2answers
45 views

How to plot a surface in maple where the range is given by an expression, not constants?

Im trying to plot the surface $z=(1+x^2)/(1+y^2)$ , but specifically the part of the surface that is above $|x|+|y|\leq1$. Cant seem to find any information on how to produce a plot in maple, where ...
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2answers
84 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
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1answer
40 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
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2answers
99 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
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1answer
102 views

Find the area of the indicated surface

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = a^2$ inside the circular cylinder $x^2 + y^2 = ay$ ($r = a\sin(\theta)$ in polar coordinates), with $a > 0$. First time posting ...
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1answer
49 views

Why is the partial derivative of a surface a curve?

I'm trying to understand the proof for Green's Theorem and I've stumbled upon a few problems. In my notes, it says that: If $E$ is a simple (flat?) surface in $\mathbb{R}^2$ (I've been trying to ...
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1answer
37 views

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 ...
3
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1answer
29 views

Proof of Castalnuovo's rationality criterion

Let $S$ be a complex projective smooth surface. If $D$ is a divisor on $S$, let's write $h^i(D)$ for $dim H^i(S,\mathcal{O}_S(D))$, where $\mathcal{O}_S(D)$ is the invertible sheaf associated to $D$. ...
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2answers
94 views

Total Curvature of 4 pi

What does it mean for a surface to have a total curvature of $4\pi $? I have seen that both the catenoid and Enneper surface are the only minimal surfaces that have this total curvature, but I don't ...
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2answers
116 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
2
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1answer
79 views

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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2answers
45 views

Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?
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1answer
93 views

Any interesting properties of Fermat's Last Theorem Surfaces?

I wonder if there are any interesting geometric (as opposed to number-theoretic) properties of what might be called Fermat's Last Theorem surfaces, i.e., $x^d + y^d = z^d$. Below are the surfaces for ...
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diagrams of twist spun torus knots

Kindly can you explain to me how to obtain the double twist spun of torus knots from tangle diagram of the given torus knot. I found the method here ...
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2answers
99 views

2D data fitting

I have some numbers as a function of 2 variables: $(x, y) \mapsto z$. I would like to know which function $z=z(x,y)$ best fits my data. Unfortunately, I don't have any hint, I mean, there's no ...
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19 views

For any diagram of a surface-knot, what are the possible connections of edges?

The singular set of a surface-knot diagram is a disjoint union of i) a graph, which has 1- and 6-valent vertices corresponding to branch points and triple points respectively. ii) circles which has no ...
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1answer
59 views

Is every quadratic surface in $\mathbb{P}^3$ ruled?

Let $\mathbb{P}$ be the projective line over an algebraically closed field $k$. Is it true that every quadratic surface in $\mathbb{P}^3$ is ruled? How can one see that this is the case? This ...
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1answer
47 views

Linear systems and rational maps

I'm following Beauville's book on Complex Algebraic Surfaces. If $D$ is a divisor on a surface $S$, we write $|D|$ for the set of all effective divisors linear equivalent to $D$ and we call it a ...
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0answers
40 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
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1answer
93 views

Flux and Gauss theorem

I have a problem; There seems to be something wrong with my understanding of gauss theorem. Let's say $F = [y ; x^2y; y^2z]$. I want to calculate the flux of $F$ going out of $$D = \{1 \le z \le 2 - ...
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1answer
65 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...
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1answer
46 views

Cohomology of the moving part of a linear system

Let $X$ be a smooth projective complex surface, $L$ a line bundle decomposed in its fixed and moving part as $|L|=F+|M|$. Intuitively, the inclusion of $|M|$ into $|L|$ yields an isomorphism ...
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1answer
35 views

Fixed components of linear systems on K3 surfaces

On a K3 surface, let $D$ be an effective divisor with $D^2\geq0$. Let $$D\sim D'+\Delta$$ be its decomposition in moving part and fixed part, respectively. Let $\Gamma$ be a prime component of ...
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1answer
40 views

Degree of blow up of a smooth projective surface

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$ and $\{x_i\}_{i \in I}$ be a finite set of closed points in $X$. Let $X'$ be the blow up of $X$ at these points. Then, $1)$ Is there a ...
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2answers
100 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
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1answer
50 views

Irreducible hypersurfaces vs irreducible polynomials

I know there exists a bijective correspondence between affine irreducible hypersurfaces and irreducible polynomials. This correspondence associates to each irreducible hypersurface ...
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1answer
291 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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1answer
27 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
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2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
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0answers
22 views

Catmull - Clark subdivision surfaces with a given border

Is there a method to decide the best mesh topology for a Catmull - Clark subdivision surface with a given border. I ask this because, when modeling with "blueprints", often we have only the ...
2
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1answer
44 views

Intersections with elliptic curves on a K3 surface

This is a fairly simple question. Suppose $E$ is an elliptic curve on a K3 surface $X$. Can we say that $E$ must intersect any curve $D\subset X$ of genus $g(D)\geq3$ ?
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92 views

K3 surfaces as complete intersections

I'm following Beauville's book "Complex Algebraic Surfaces". If $S$ is a K3 surface and $C$ is a smooth not hyperelliptic curve of genus g, then we have a birational morphism $\phi : ...
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1answer
50 views

Cubic surface and birational equivalence

Following Shafarevich "Algebraic Geometry II", I found this example. Let $X_3\subset\mathbb{P}^3$ a smooth cubic surface. To prove that $X_3$ is rational he claims that there is a birational map ...
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1answer
71 views

Show that space of mxm matrices form a surface

How can we show that the space of $m ~ \mathbb{x}~ m$ matrices with determinant equal to one form a $C^1$ surface of dimension $m^2-1$ in $\mathbb{R^{m^2}}$? I know there is probably a catch to it, ...