1
vote
1answer
15 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $z = a(\frac{x}{y} + \frac{y}{x})$, for constant $a \neq 0$. I guess that what was meant by that statement is that surface $S$ can be ...
1
vote
0answers
23 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 ...
3
votes
1answer
41 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 ...
2
votes
1answer
41 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 ...
1
vote
1answer
51 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 ...
1
vote
1answer
20 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 ...
1
vote
0answers
24 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 ...
1
vote
1answer
24 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 ...
1
vote
1answer
28 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 ...
0
votes
0answers
26 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
25 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 ...
3
votes
1answer
99 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; $E=L=1$ $F=M=0$ $N=G=\cos^2 u$ ...
0
votes
0answers
32 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.
2
votes
0answers
44 views

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 ...
0
votes
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...
1
vote
1answer
38 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 ...
0
votes
1answer
43 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 ...
1
vote
2answers
59 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?
2
votes
1answer
28 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 ...
2
votes
1answer
55 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 ...
0
votes
0answers
63 views

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 ...
4
votes
1answer
128 views

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 ...
20
votes
3answers
231 views

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? ...
1
vote
1answer
55 views

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 ...
2
votes
0answers
27 views

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 ...
1
vote
1answer
35 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
1
vote
0answers
57 views

Mathematically explaining a trapped surface?

I'm currently writing my thesis, and the general area is that of minimal surfaces. I have a deep interest in cosmology so have directed it towards space-time, ie applying minimal surfaces in space, ...
2
votes
0answers
59 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
37
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
3
votes
2answers
133 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
0
votes
1answer
44 views

Prove that $\textbf{II}_p\equiv0$ on $M:=f(U)$ if and only if $M$ is contained in a plane

Let $U\subset\mathbb R^2$ be open and connected, and let $\ f:U\rightarrow\mathbb R^3$ be the parametrization of a regular surface. Prove that $\textbf{II}_p\equiv0$ on $M:=f(U)$ if and only if $M$ ...
1
vote
1answer
53 views

Proof that the infinite cylinder is a regular surface.

I have to proof that the circular cylinder $M=\{(x,y,z)\in\mathbb{R}^3\mid x^2 + y^2 = r^2\}$ is a regular surface, where $r$ is a constant, $r>0$. Then I have to see also that $\mathrm x\colon ...
0
votes
1answer
174 views

compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...
0
votes
1answer
228 views

Outer Unit Normal: Cylinder

I have a cylinder occupying the region $x_{1}^{2}+x_{2}^{2} = R^2$ and $-G< x_3 < 0$ All I want to do is define the outer unit normal on the curved face. I thought about just calling it $e_1$ ...
4
votes
1answer
109 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
2
votes
1answer
59 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
1
vote
0answers
88 views

Surface orientation

Let $S_{1}$ and $S_{2}$ be two oriented surfaces ($N_{1}$ and $N_{2}$ their normal fields, respectively). We say that a local diffeomorphism $f$ : $S_{1}$$\rightarrow$$S_{2}$ preserves orientation if ...
3
votes
1answer
46 views

What does it mean for a surface to evolve with divergence-free velocity?

Suppose we have an evolving hypersurface which evolves with a velocity field $V$, such that $\nabla_S \cdot V = 0$ where $\nabla_S$ is the surface or tangential gradient. What does this mean? What ...
1
vote
1answer
58 views

The equation of a surface created by the extrusion of a 2D closed curve along a path

How do I obtain the equation of a surface created by the extrusion of a circle (or ellipse) created on the XY plane along a parabola or a parametric curve which lies on the YZ plane. The goal is to ...
2
votes
1answer
112 views

Flowlines of blobs

I have the following formula for blobs/metaballs, which is said to be the same as the one used for electromagnetism: \begin{gather} f(x,y,z) = \frac{d(A,B)}{\sqrt{(x-xA)^2 + (y-yA)^2 + (z-zA)^2}} \\ ...
0
votes
0answers
52 views

Rheotomic surfaces parameterization?

Are there parameterizations for rheotomic surfaces? Or, am I stuck with implicit formulas and marching cubes for plotting points? Are there special cases where the surfaces are parameterizable? Here ...
1
vote
1answer
217 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
1
vote
1answer
108 views

Does a Möbius strip have only one shape? Or may it have different shapes?

I'm reading a book about geometry, and after thinking and viewing the Möbius strip, I want to know whether the book is right or not. The book says with a little description (that I can't write here ...
1
vote
0answers
154 views

$N$ equally spaced points on an ellipsoid

I would like to find a algorithm for determining the $(x,y,z)$ co-ordinates for evenly distributed $N$ points on the surface of an ellipsoid. These points must be spaced from its nearest neighbour ...
1
vote
1answer
34 views

Formula for curvature of two intersecting surfaces in terms of their normal curvature.

I have been privately reading DoCarmo recently, and have been attempting to do some of the problems. I am stuck on this one, it is problem 14 in section 3.2 for those interested. If someone could show ...
0
votes
0answers
38 views

Surface Reconstruction from Hessian Field

I am looking for references regarding surface reconstruction. Consider a point cloud in $\mathbb{R}^3$ with the Hessian (or possibly second fundamental form) defined at each point. I would like to ...
10
votes
1answer
144 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
1
vote
1answer
248 views

Surface area element of an ellipsoid

I would like to evaluate an integral numerically over the surface of an ellipsoid. Take an $N \times N$ grid over the parameter space $(u, v) \in [0, 2\pi) \times [0, \pi) $. A simple approximation of ...
3
votes
2answers
172 views

hypersurface evolving with tangential velocity

If a hypersurface $S_t$ evolves with velocity only in the tangential direction, is $S_t \equiv S_0$ for all $S$? This is what I have read is true (or something very similar). Can someone give me an ...