2
votes
1answer
34 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
34 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
22 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
41 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
0answers
34 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 ...
2
votes
2answers
60 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
32 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
0answers
44 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
34 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 ...
1
vote
2answers
71 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
2answers
58 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
42 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 ...
-1
votes
1answer
46 views

Calculate the curvature of a space curve at the point M(-1,5,-4)

In order to calculate the curvature for this space curve, do I use this formula? And where does the point $M$ come into this? P.S. This is probably a silly question, but I'm new to differential ...
1
vote
1answer
71 views

Radius of curvature for the plane curve $x^3 + y^3 = 12xy$.

Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.
2
votes
1answer
91 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
1
vote
1answer
60 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
7
votes
2answers
106 views

Where does this expression of Gaussian curvature come from?

In my Differential Geometry course, we have seen a way to calculate the Gaussian curvature $K$ given a metric expressed as the sum of two Pfaff forms $Q = ω_1^2 + ω_2^2$: we find another Pfaff form ...
1
vote
2answers
63 views

Is a tangent to a curve in a hyperbolic plane straight?

Consider a projective plane with an absolute quadric, so that it is a hyperbolic plane. Given a curve I wonder how the tangent to a curve is defined in a plane with constant positive curvature. I ...
1
vote
1answer
44 views

Gaussian curvature of a parallel surface

Question 11 section 3.5 in Do carmo part c.Let a surfae x have constant mean curvature equal to c does not equal 0 and consider the parallel surface to x at a distance 1/2c. Prove that this parallel ...
3
votes
2answers
65 views

Computing the Gaussian curvature of this surface $z=e^{(-1/2)(x^2+y^2)}$.

Compute the Gaussian curvature of $z=e^{(-1/2)(x^2+y^2)}$. Sketch this surface and show where $K=0 $, $K>0$, and $K<0$. So would the easiest way to do this question be to construct a ...
0
votes
1answer
20 views

Mean curvature an asymptotic directions

I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two ...
1
vote
2answers
61 views

Gauss Curvature of a Surface

Find the Gauss curvature of a surface with the first quadratic form: $$\mathrm{d}s^2 = \mathrm{d}u^2 + 2\cos a(u,v)\mathrm{d}u\,\mathrm{d}v + \mathrm{d}v^2.$$ I have found $E$, $F$, and $G$. $E = ...
2
votes
0answers
42 views

Curvatures of 4-Dimensional Parallel Curve

Let $\vec{\alpha}: I \rightarrow \mathbb{R}^4$ be an arc-length parametrized curve in $\mathbb{R}^4$ with curvatures $k_1, k_2, k_3$. The principal normal unit vector is $\vec{n} = ...
1
vote
1answer
53 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 $k_2$, ...
1
vote
0answers
27 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
2
votes
0answers
40 views

Frenet formulas for curves in arbitrary Riemann manifold

As far as I understand, to have Frenet formulas one would need a curve, embedded in $\mathbb{R^n}$ and, desirably, naturally parametized. But there are homonomic notions of curvature and torsion of ...
1
vote
1answer
37 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 ...
2
votes
3answers
93 views

Finding the Total Curvature of Plane Curves

I'm trying to find the total curvature (or equivalently, rotation index, winding number etc.) of a plane curve (closed plane curves) given by $$\gamma(t)=(\cos(t),\sin(nt)), 0\leq t\leq 2\pi$$for each ...
1
vote
0answers
52 views

Proof check for critical point definition with mean curvature

I'm currently trying to prove: "Definition: We say that a surface $S \in R^3$ is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ...
1
vote
1answer
55 views

Cartan formalism calculation

Just to test out the Cartan formalism, I decided to apply it to the sphere. So, it admits a metric, $$\mathrm{d}s^2 = \mathrm{d}r^2 + r^2 \sin^2 \phi \mathrm{d}\theta^2 + r^2 \mathrm{d}\phi^2$$ from ...
1
vote
2answers
87 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?
3
votes
1answer
54 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
3
votes
1answer
55 views

Curvature of saddle by definition

I'm trying to compute the principle curvatures of the saddle $M$ defined by $z= y^2 -x^2$ at the point $p = (0,0,0)$, but I know my computations are wrong. Maybe you can help to see where I went ...
1
vote
0answers
51 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
7
votes
1answer
120 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
0
votes
0answers
22 views

How can I retreive Cartesian positions from curvature and torsion?

I have a set of 3D points in the Cartesian space. What I want is to extract the TNB (frenet frames) of these points together with the curvature and torsion. Which seems to be easy. Then I want to do ...
1
vote
1answer
309 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
0
votes
0answers
87 views

Question about spherical curvature ( binormal, tangent vectors)

Let $a:I\mapsto R^3$ be a unit speed curve. if $\rho^2+(\rho'\sigma)^2$ is constant and equal to $r^2$, show that a curve is on a sphere with r radius. answer is: define $\gamma:I\mapsto R^3$ ...
2
votes
1answer
97 views

Trying to understand Gauss-Bonnet theorem

Update: I was able to figure this out on my own. The problem was that I assumed the $\phi$ coordinate runs from $0$ to $2\pi$, but the metric so defined has a conical singularity. To eliminate the ...
1
vote
0answers
134 views

The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
4
votes
2answers
130 views

when can you estimate curvature from finite information about two geodesics?

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
0
votes
1answer
44 views

gaussian curvature in isotherm parameterization

Consider an isothermic parameterization with metric $$(g_{ij}) = \begin{bmatrix} \lambda^2 & 0 \\ 0 & \lambda^2\end{bmatrix}$$ with $\lambda = \lambda(u^1,u^2)>0$. Then the gaussian ...
10
votes
2answers
289 views

Ricci curvature: step in proof of a paper by Hamilton

In Hamilton's paper "The Ricci Curvature Equation" (in Seminar on Nonlinear Partial Differential Equations, here), I can do all of Lemma 4.2 except for the following relation: $$ ...
2
votes
1answer
33 views

If $dX_1 = dX_2$ then curvatures of $\nabla^{X_1}$ and $\nabla^{X_2}$ agree

Let $E \simeq M \times \mathbb C$ be a trivial smooth complex line bundle over the Riemann surface $M$ and let $S \colon M \to E$ be its smooth nowhere vanishing section. Let $\nabla^1$ and $\nabla^2$ ...
0
votes
1answer
36 views

Let $c:I\rightarrow\mathbb R^3$ a regularly parametrized curve with curvature $\kappa=0$

where $I\subset\mathbb R$ is an interval. Show that the image of $c$ is contained in a straight line. we defined the curvature with; $\kappa_c(t)= ...
1
vote
2answers
86 views

Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
0
votes
1answer
133 views

Differential geometry - proving an expression for the principal curvature

I have tried to solve the following problem for some time but cannot get it right. Let $X: U \rightarrow \mathbb{R}^{3}$ be a regular parametrized surface in $\mathbb{R}^{3}$ with Gauss map $ N: ...
0
votes
1answer
168 views

Gaussian curvature and mean curvature of an ellipsoid [duplicate]

I need help with this differential geometry problem dealing with Gaussian curvature and mean curvature Given an ellipsoid with parametric representation $(a\cos u\cos v, b\cos u \sin v, c\sin u)$ ...
3
votes
0answers
50 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
1
vote
1answer
171 views

gaussian mean curvature

I am trying to review, and learn about how to compute and gaussian and mean curvature. Given $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, how can I compute the gaussian and mean ...