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4
votes
1answer
45 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
1
vote
0answers
17 views

Embedding of manifolds of constant negative curvature

Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
1
vote
1answer
109 views

Curvature of curve not parametrized by arclength

If I have a curve that is not parametrized by arclength, is the curvature still $||\gamma''(t)||$? I am not so sure about this, cause then we don't know that $\gamma'' \perp \gamma'$ holds, so the ...
3
votes
2answers
31 views

Number of Curvature Maxima of a 2D Cubic Bezier curve

I am trying to prove that a standard cubic Bezier curve can only have at most 2 curvature maxima over $t \in [0,1]$. Assuming that no 3 adjacent control points are colinear, the curvature will either ...
3
votes
1answer
49 views

Reparametrize by arc length and find the signed curvature

Let $$g(t)=(e^t \cos(t),e^t \sin(t))$$ Reparametrize $g(t)$ by arc length starting at $0$, then find the signed curvature of the unit-speed reparametization. I found that the ...
1
vote
1answer
65 views

What are some interesting topics for a differential geometry essay?

As an assignment I have been asked to write an essay on an aspect related to differential geometry. My lecturer has proposed topics such as architecture, navigation and computer graphics. I personally ...
2
votes
1answer
46 views

Curvature kappa with known acceleration, unit normal and unit tangent vectors

The acceleration of a particle is $a(t) = (4\sin t \cos t)T(t) + (4e^t \sin^2 (t/6))N(t)$, where $T(t)$ is the unit tangent vector and $N(t)$ is the unit normal vector. At $t = \pi/2$, the speed of ...
3
votes
0answers
45 views

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
1
vote
0answers
17 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
2
votes
1answer
51 views

Curvature in complex analysis

Apparently, the curvature of the image contour of the unit circle $|z| = 1$ under the conformal mapping $w = f(z)$ is $$\frac{1}{|zf'(z)|} \text{Re}\bigg(1 + \frac{zf''(z)}{f'(z)}\bigg).$$ How does ...
0
votes
1answer
23 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
0
votes
0answers
25 views

radius of curvature problem

Find the radius of curvature at $\theta$ on the curve x = a log$(cot \theta/2 - cos \theta)$ y = a $sin \theta$ Can you please help me on how to approach this problem ?
1
vote
0answers
18 views

A non-rigorous explanation of curvature of curves in 3-space

In a second year calculus class there is usually the definition of curvature as $$\kappa=\left|\frac{d\bf{T}}{ds}\right|$$where $\bf{T}\rm(t)=\frac{r^\prime(t)}{|r^\prime(t)|}$ is the unit tangent ...
0
votes
0answers
19 views

Point of maximum curvature

I'm trying to find the point where the change in y becomes more or less independent of the change in x in this equation: $$y=7\times 10^{-5}\exp\left(\dfrac{-x}{3.0453\times 10^{-7}}\right)+9.1\times ...
0
votes
1answer
38 views

Find point where radius of curvature is minimum

Find the point where radius of curvature is minimum for the curve $$x^2y=a\left(x^2+\frac{a^2}{\sqrt{5}}\right)$$
0
votes
1answer
123 views

Proof of curvature of a curve described by Polar Coordinates

I have been looking everywhere for a proof on the curvature of a plane curve that is represented in polar coordinates. I am close in proving it myself, however, I seem to be missing a particular part ...
1
vote
1answer
102 views

Prove that the line is tangent to the curve at the point.

Hello can someone please walk me through part a and b of the below question? I really want to understand it but am having a hard time figuring out the solution. I know how to calculate curvature for a ...
0
votes
0answers
37 views

A converse proof that involves Torsion, curvature, and differentiation that equates to 0

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
1
vote
1answer
57 views

A proof that involves Torsion, curvature, and differentiation that equates to 0

I know that I am supposed to write alpha = lambda*T + mu*N + v*B then differentiate then use the fact that {T,N,B} is a basis in R^3. I am just unsure how to write alpha as a combination using the ...
0
votes
2answers
90 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
1
vote
2answers
52 views

Tractricoid as a pseudosphere (surface with constant negative curvature)

How to motivate/calculate/prove see that the tractricoid, i.e. a tractrix rotated about its asymptote, has a constant negative curvature? What are the hyperbolic lines on a tractricoid and how to see ...
0
votes
0answers
33 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
4
votes
0answers
36 views

Non-vanishing differential forms and flatness

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...
0
votes
0answers
12 views

How to draw a border at a specific distance from a cylinder outline

I have a small cylinder (Cylinder A) with its minor radius of A. Minor radius measures the minor radius of the cylinder ellipse. I need to draw a border at a distance of X from the border of ...
2
votes
1answer
42 views

Monotonic curvature and self intersections.

I'm trying to prove that if $\alpha: I \to \Bbb R^2$ is a differentiable curve, where $I$ is an interval, has strictly monotonic curvature, then $\alpha$ has no self-intersections. My attempt: We ...
2
votes
1answer
45 views

The curvature of a Cycloid at its cusps.

My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps. Having left it as an open problem, I thought it'd be ...
1
vote
0answers
31 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
0
votes
0answers
67 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
1
vote
1answer
56 views

Which axiom makes a vector space flat?

First of all, I'm not sure if this question even makes sense, i.e. is there a notion of curvature on a vector space structure. However, when dealing with vector spaces (here I am mostly thinking of ...
1
vote
1answer
26 views

An assumption used to derive the curvature tensor for Riemannian submersions

I was reading the literature about Riemannian submersions, and I came across the result showing the relation between the curvature tensor $\bar{R}$ in a manifold $M$ and the curvature tensor $R$ in a ...
1
vote
0answers
45 views

curvature of a plane curve

I'm trying to prove the formula to calculate the curvature of a plane curve. But I end up with the wrong sign and can't figure out why: I want to proof $\kappa(t) = \frac{\dot c(t) \cdot \ddot ...
3
votes
0answers
28 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
2
votes
2answers
72 views

Ricci Tensor, Curvature and Scalar Curvature computation from definition

I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ...
0
votes
1answer
37 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
1
vote
2answers
33 views

How to proof curvature and torsion are independent

As we know that curvature describes the change of curve in tangent and normal plane, while torsion describes the change the curve in binomial and normal plane. Assume we have a trajectory with length ...
1
vote
1answer
57 views

Definition of CAT(0) metric space

I have a question regarding the definition of CAT(0) spaces. I am using the following definition: $X$ complete metric space is CAT(0) if $\forall z,y \in X$, $\exists m \in X$ such that $\forall x ...
1
vote
0answers
50 views

Shape operator and orthogonality of eigenvectors

When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ...
1
vote
0answers
39 views

Finding curve that minimizes an integral due to constraints

In the euclidean plane I want a smooth curve $\gamma (t)$ which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma ...
0
votes
1answer
71 views

maximum curvature of 2D Cubic Bezier

Given a 2D cubic Bezier segment defined by P0, P1, P2, P3, here's what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a ...
0
votes
1answer
49 views

Quasi-concavity of a function of two variables such as $z=(x^a + y^b)^2$

If I have a function such as $z=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
0
votes
0answers
15 views

point of maximal curvature of a vector of discrete values

How can I approximate the point of maximal curvature given a vector of points rather than a function? I do find the inflection point(s) by comparing the slope within a small window before and after a ...
1
vote
2answers
104 views

Global and local coordinates on a manifold, and their relations to curvature

I would be pleased to have some information about coordinates in differential geometry. A) First I would like to check whether or not the definitions I use are correct. (Mainly for the sake of ...
1
vote
0answers
21 views

Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold

I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$ where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian ...
1
vote
1answer
67 views

Resample Bézier Curve with curvature and number of points constraints

I have an algorithm that implements an uniform resample process throughout a Bézier curve. This is done using a chord parametrization process. However, the results achieved do not accomplish my ...
1
vote
2answers
72 views

Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
0
votes
1answer
44 views

Proving that a surface is isometric to the plane

A surface $S$ has first fundamental form $du^2 + G(u,v)dv^2$ and curvature $0$. Also the curve $u=0$ is a geodesic when parametrized by arclength. Prove that $G(u,v) = 1$ i.e. that $S$ is isometric ...
4
votes
2answers
61 views

What curvature conditions make a surface rigid?

Consider a compact surface $S$, possibly with boundary, embedded in $\mathbb{R}^3$, with the induced Riemannian metric. I believe that if $S$ has constant positive Gaussian curvature (that is, $S$ is ...
1
vote
1answer
52 views

On Constant mean curvature surfaces.

I have two involved questions, firstly, I know that the gauss map sends a surface to the unit sphere, so for a surface $\Sigma\subset\Bbb R^3$, parametrised by $u:U\subset\Bbb R^2\to \Bbb R^3$. Would ...
1
vote
1answer
52 views

Show that total curvature of ellipse is $2\pi$

I'm trying to show that the total curvature $$K=\int_C\kappa\,ds$$ is equal to $2\pi$ over the ellipse $C$ with axes $a,b$ (and $\kappa$ is curvature). I computed: $$x(t)=(a\cos t,b\sin t,0) \\ ...
0
votes
1answer
20 views

Estimating curvature of oscillatory curve based on global constraints

I have a heuristic question about using global constraints of a problem to make estimates of local features of a curve, such as its curvature. Consider a suitably well behaved function on ...