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2
votes
1answer
27 views

The curvatures of a transformed surface under a similarity transformation

Setup: Let $f:\mathbb R^3\to\mathbb R^3$ be a similarity transformation. Then $f=rA+b$ for some fixed orthogonal matrix $A$, vector $b$ and nonzero real $r$. Suppose $S$ is a surface, and $S'=f(S)$. ...
3
votes
0answers
18 views

Total curvature of an ovaloid.

I have the following exercise that I have to solve without using Gauss-Bonnet theorem. We say that a compact surface $\Sigma \subset \mathbb{R}^3$ is an ovaloid if the Gaussian curvature $K(p)>0$ ...
0
votes
0answers
30 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
1
vote
0answers
20 views

Finding the “elbow” in a set of numbers

I have point X on a map. I have other points also (call them A, B, C and so), and I know how far away they are from point X. For example: A: 1 unit; B: 2 units; C: 2 units; D: 9 units. I want to ...
4
votes
1answer
31 views

Characterize the sphere using mean curvature.

We know the following result: if $\Sigma$ is a compact surface than $$ \int_{\Sigma}H^2 \ge 4 \pi, $$ where with $H= \frac{1}{2}(\kappa_1+\kappa_2)$ we denote the main curvature. I have to prove that ...
2
votes
0answers
36 views

Geometric interpretation of Gaussian curvature.

We have the following result from Do Carmo book of differential Geometry: "Let $p$ be a point of a surface $\Sigma$ such that the Gaussian curvature $K(p) \neq 0$, and let $V$ be a connected ...
2
votes
1answer
46 views

Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let two $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with ...
1
vote
2answers
28 views

Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
1
vote
0answers
8 views

Hermite Spline curvature

I have a doubt about the Hermite Spline. Is it the interpolation with the minimum value of curvature among the possible interpolative functions between two points? Is it possible to demonstrate it? ...
0
votes
1answer
26 views

Differential Geometry - normal curvature of an embedded torus in the direction of a given tangent vector

Given the parametrization of an embedded torus as: sigma(u,v) = ((2+cos(u))cos(v), (2+cos(u)sin(v), sin(u)) and that point p: (1,0,0) is a point on the torus. I need to calculate two things: a. The ...
1
vote
1answer
18 views

Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
2
votes
0answers
37 views

Find the curve. Differential Geometry.

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. ¿Any ...
7
votes
3answers
317 views

Why is the Riemann curvature tensor the technical expression of curvature?

According to my textbook on general relativity (Sean Carrol's book) and differential geometry, the Reimann curvature tensor is the technical expression of curvature. What makes the tensor so special? ...
1
vote
0answers
40 views

Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
2
votes
1answer
23 views

Question on calculating curvature of a surface given implicitly

I want to find, as an exercise, an expression for the curvature of a surface given by the zero set of a function. I reached a final expression, but when I test it for a sphere I get a non-constant ...
0
votes
1answer
30 views

A point of infinite curvature on a curve

Let $\gamma(t)$ be a $C^r$ smooth curve in the plane. Suppose $r\ge 2$, so that one can define the curvature $\kappa(t)$ at $\gamma(t)$. For example, $\kappa(t)=0$ means that the curve is kind of flat ...
1
vote
1answer
35 views

Using Principal Directions and Curvatures to Find Point On Surface

Given the principal directions (max and min), principal curvatures, and normal of a surface at point n, how would you go about looking for a point on the surface at a given vector distance from n? ...
1
vote
1answer
20 views

Is there a name for the coordinate on a function which has maximum curvature?

I found out how to find the maximum curvature, by differentiating the curve function. I am wondering if there is a mathematical term, or if there isn't one what is the most elegant way to represent ...
2
votes
1answer
37 views

Vanishing of the Riemann tensor

The Riemann tensor in a coordinate basis is $$R^{i}_{\,jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^m_{jl}\Gamma^i_{mk} - \Gamma^m_{jk}\Gamma^i_{ml}$$ Consider $\mathbb{R}^2$ ...
0
votes
0answers
27 views

Mean curvature of polar parametric surface

For the purposes of modelling a fluid mechanics experiment, I'm dealing with a convex surface parametrized by the azimuth $\theta$ and an arc length $s$ along the surface. The points on the surface ...
0
votes
0answers
14 views

confirmation of curvature of paramteric curve

I just want to confirm the following calculation for curvature of a parametric curve: Given parametric curve $r(t) = (5 \sin t, 5 \sin t, 3 \cos t) $, I want to confirm that the curvature is given as ...
3
votes
2answers
73 views

Zero Sectional Curvature implies exp is a local isometry

Im studying DoCarmo's book Riemannian Geometry, the first problem of the chapter 5 (Jacobi Fields) states that If $(M,g)$ is a riemannian manifold with sectional curvature identically zero, show that ...
0
votes
0answers
31 views

Understanding Euler density

I know the definition of Euler density in terms of antisymetrized contractions of products of the Riemann curvature tensor, ie Euler density is the $\mathcal{R}^n$ in these formulae: And I know ...
0
votes
0answers
18 views

Finding inflection points and concavity for given function

I have to explore function $\frac{2x-4}{1+x^2}$ It contains finding definition area of function, finding roots etc. One of the points is to find inflection points and concavity of function. I cannot ...
1
vote
2answers
45 views

If $0 < \theta < \frac{\pi}{2}$, then $\gamma$ is a logarithmic spiral

Let $\gamma$ be a plane curve parametrized by the arc length, having the property that its tangent vector $T(t)$ forms a fixed angle $\theta$ with $\gamma(t)$. Before explaining where I am, let ...
1
vote
1answer
73 views

Parametrization where coordinates lines are lines of curvature

I am asked to prove that given a surface $S$ and a point $p\in S$ non-umbilical, then there exists $U$ open in $\mathbb{R}^2$, there exists $Y:U\subset \mathbb{R}^2\longrightarrow \mathbb{R}^3$ a ...
0
votes
0answers
58 views

Curvature and Pfaffian forms in terms of the Riemann tensor

I am teaching my self differential geometry, but I am mainly familiar with classic tensor notation. In modern Cartan exterior form notation the curvature forn $\Omega$ and the Pfaffian seem to do the ...
1
vote
2answers
65 views

Hyperbolic geometry when the curvature is constant and negative but not -1

Help I am getting completely confused Hyperbolic geometry is the geometry of surfaces of a constant negative Gaussian curvature, in most formula's it is almost assumed this constant negative ...
0
votes
1answer
64 views

Radius vs Radius of curvature of an ellipse

I am a bit confused by the physical meaning of radius vs radius of curvature, with regards to an ellipse. For a standard ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ In this case, the $a$ ...
0
votes
1answer
32 views

preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
1
vote
0answers
35 views

Does covariant derivative commute with “generalized contraction”_ About the proof of 2nd Bianchi identity

I am reading the proof of second Bianchi identity on wiki. In the proof, it says the following condition must satisfy: $$((D_X R) (Y,Z)) (W) + R (D_XY,Z) W + R(Y,D_XZ) W + R(Y,Z) D_X W = D_X ...
1
vote
0answers
32 views

Prove the local expression of Riemannian curvature tensor

I try to prove the following expression of Riemannian curvature tensor: For local coordinate $\{x^i\}$, let $g_{ij}=g(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})$ and ...
1
vote
0answers
12 views

Linearlized curvature operator

While reading a paper, I came across the term for linearized curvature operator \begin{eqnarray} \kappa_1 = -\frac{1}{{(1+x^2)^{\frac{3}{2}}}}\frac{\,d }{\,d x^2} + ...
0
votes
1answer
32 views

Let $\gamma: (\alpha, \beta) \rightarrow \mathbb R^3$ be a space curve with speed $1$ s.t $|| \gamma(s) || = R > 0$. Show $\kappa(s) \ge \frac 1 R$.

Let $\gamma: (\alpha, \beta) \rightarrow \mathbb R^3$ be a space curve with speed $1$ s.t $|| \gamma(s) || = R > 0$. I want to show that the curvature $\kappa(s) \ge \frac 1 R$ for all $s \in ...
1
vote
1answer
34 views

Failure to close of horizontal lift on principal bundles.

When considering the geometrical meaning of the curvature on principal bundle $\pi: P \rightarrow M$, let us consider a coordinate system ${x_μ}$ on a chart $U$. Let $V = ∂/∂_{x_1}$ and $W = ...
2
votes
0answers
114 views

Geometric meaning of $\nabla_{[i}(x^i \nabla_{j]}x^j)$ and $(\nabla_{[i}x^i )\nabla_{j]}x^j$

While teaching myself tensor calculus I have come up with [this] http://mathematica.stackexchange.com/a/71613/12306 {The proof of the 2-D hairy ball theorem). When trying to generalize this proof ...
1
vote
1answer
47 views

Computation of the extrinsic curvature tensor for a warp drive metric.

In Miguel Alcubierre's renowned paper discussing a "warp drive" metric, he discusses the extrinsic curvature. Here is an extract. My questions are quite trivial to someone who understands the ...
1
vote
1answer
79 views

Visualizing Ricci scalar curvature

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
1
vote
3answers
64 views

Numerical Approximation for 2D Curvature

I have a list of points (x, y) that are taken from an unknown 2D parametric curve $\vec{f}(t)$. These points are monotonically increasing in t (ie: they're a "connect the dots" version of ...
0
votes
0answers
29 views

Sectional curvature in 3-dimensions

I wonder how to compute the sectional curvature of 3-dimensional objects eg. unit ball, $H=\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb{R}^{4}:x_{0}^{2}-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=1$ and ...
0
votes
2answers
56 views

Prove that $\frac{dN}{ds}=-\kappa T$

Prove that $\frac{dN}{ds}=-\kappa T$, where $N$ is the oriented normal, and $T$ is the unit tangent vector, and $s$ is arc-length parameter. Here's what I've got so far from my note and I don't ...
0
votes
0answers
9 views

compute next frame curvature torsion based on existing tangent, normal, binormal, position, curvature and torsion

I have a question that is making me headache. Suppose I have $r_{s}$, $T_{s}$, $N_{s}$, $B_{s}$, $\kappa_{s}$ and $\tau_{s}$ for position of the sample, tangent, normal, binormal, curvature and ...
1
vote
2answers
54 views

Using eigenvalues of a hessian matrix vs D operation to classify critical points.

Having recently covered using the discriminant, $D(x_0,y_0)$, for classifying critical points of equations of two variables. For example: $$R(x,y)=-x^2+4x+2xy+8y-2y^2$$ to find that $(6,8)$ is the ...
1
vote
1answer
39 views

Discrete Gauß and geodesic curvature

Imagine that you have an n-polygon $S$ and you wanted to calculated the discrete Gaussian or gedoesic curvature. How are they defined? If $p$ is a vertex of $S$ then Gauß-Bonnet suggests that the ...
0
votes
1answer
74 views

How to find the sum of maximum and minimum curvature in an ellipse?

I am having difficulty finding the sum of maximum and minimum curvature of the ellipse $9(x-1)^2 + y^2 = 9$. I know that I am supposed to parametrize the ellipse as $f(x(t), y(t))$, with $x(t) = 1 + ...
0
votes
1answer
32 views

Negative Gauss Curvature

Let S be a manifold of dimension 2, compact and orientable. Suppose its border is made of k geodesic circumferences, with $k \geq 3$. Show that there exists a point in S with negative Gauss ...
0
votes
1answer
75 views

Christoffel symbols of a surface of revolution

I am looking for a way to write down the Christoffel symbols for a surface of revolution. They are given by ...
1
vote
0answers
52 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
5
votes
2answers
74 views

What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
0
votes
0answers
22 views

sum of two curvatures

I have a question about sum of two relative curvature and torsion. The problem is as follow. Suppose we have three points, Ps, ...