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Questions tagged [curvature]

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

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Computing the shape operator $S_\eta :T_pN \rightarrow T_pN$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by $$f(x, y) = (\text{cos}(x)(\text{cos}(y)+4), \text{sin}(x)(\text{cos}(y) +4), \text{sin}(y)).$$ How would I go about computing the shape ...
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How to compute $B(X, Y) = \bar{\nabla}_\bar{X}\bar{Y} - \nabla_XY$

Let $g : \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be the immersion defined by $$g(x, y) = (cos(x), sin(x), cos(y), sin(y)).$$ Let $e_1 = \frac{\partial}{\partial x}$ and $e_2 = \frac{\partial}{\partial ...
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Is the curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the $\textit{exterior ...
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How to calculate the geodesic curvature of a discrete 3D curve?

I have coordinates of a set of points that form a closed loop that lies in a 3D surface. I know the equation of the surface and I can calculate it's surface normal at any point. I found that for a ...
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Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $\omega$ on a principal bundle and its curvature form $\Omega$, I can state that $\Omega=d_\omega\omega$ alike I do in the ...
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Get points in the plane of an Euler spiral given by curvature

I am currently working on a visualization of a road network. I am using OpenDrive as my standard for road description. I now have a problem with visualizing the curved parts of roads. These are given ...
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Finding curvature of a surface of revolution with metric inherited from $\mathbb{R^3}$

I have a curve in $\mathbb{R^3}$ as $z= f(x)$ in the $xz$- plane and we let $S$ be the surface generated from this curve in the space by revolving it about the $z$- axis. Now, I'm asked to find it's ...
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Gaussian curvature with Laplacian

In a lot of papers and books, I have seen the following expression of Gauss Curvature in $2$-dimensional surfaces with a conformal metric $$\overline{g} = e^{2u}g$$ $$K - \overline{K} e^{2u} = \...
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Weitzenböck identity for $TM$-valued differential forms

Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$: The ...
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Can someone help in computing curvature tensor of a surface?

I have a problem, I've been thinking about all day. Came across this while browsing some lecture notes online. So, I have a surface in space say, described as $z= f(x,y)$ and I want to find it's ...
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Does there exist an opposite to curve length in integral calculus, “radius length”?

Consider the formula for curve (arc-) length in integral calculus: $$\int_a^b\sqrt{1+\left(\frac{\partial y}{\partial x}\right)^2}dx$$ What would the geometrical opposite thing to measure be? Some ...
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Curvature of the moment curve

I would like to compute the curvature of the moment curve in $\mathbb R^n$: $$ (s,s^2,s^3,\dots,s^n). $$ I am especially interested in the curvature at $0$. But this is not parametrized by arc length. ...
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Definitions of Gaussian curvature

In my differential geometry course the professor has defined Gaussian curvature to be the determinant of the shape operator, $$K=\det(S(p))$$ However most books that I have been following along with ...
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How to prove that Frenet frame is independent of the choices of parameters?

When I am reading ''A course in differential geometry'' of Klingenberg, I cannot be sure the Frenet frame defined in this book is independent of the choice of parameter of a curve. As a result, the ...
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Why the constant curvature curves on plane don't contains general hyperbola?

For a curve $y=f(x)$, the curvature is $$ k=\frac{|y''|}{(1+y'^2)^{3/2}} $$ If the curvature is a constant, there are 3 cases \begin{align} \frac{y''}{(1+y'^2)^{3/2}}=0 \tag{1}\\ \frac{y''}{(1+y'^2)^...
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Finding Radius of Curvature

I am not very clear on how to find the radius of curvature or the radius of the arc of circle formed when we are given some other parameters, say the velocity of a moving vehicle and the angle at ...
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Visualizing the Lie Bracket in connection with Torsion

I have seen this kind of picture a lot, linking the Lie Bracket with the Torsion (e.g. 1, 2, 3, 4). I will report for convenience one of such picture, from Hehl and Obukhov review article For some ...
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Confusion in deriving curvature and torsion formulae for general curves

I've looked at this answer and this answer , in addition to the MathWorld articles for curvature and torsion, but for the life of me cannot derive the formulae for general curves by myself. My issue ...
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Why do the principle curvatures provide the maximum and minimum values of the normal curvature of a curve?

I am having difficulty showing that the principle curvatures maximise and minimise the normal curvature of a given curve, say $\alpha(t) \in S$ where $S$ is a regular surface. Given $T$ is the unit ...
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Non-flat connection on trivial bundle?

From what I have read it seems like there exists non-flat connections on trivial vector/principal bundles. However I can't find any notes on it or examples. Can anyone confirm that such connections ...
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The Four-Vertex Theorem and the existence of curvature

All statements of the four-vertex theorem I've seen so far talk about simple, closed, smooth plane curves. And the theorem itself is about the (extrema of the) curvature of these curves. What I ...
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Second Fundamental Form, Principle Curvature and Tangent Plane Vectors Orthogonality

As I'm studying differential geometry of surfaces, I encountered the possibility of the first fundamental form $F_{I}$ being an identity matrix making the shape operator $S=F_{I}^{-1}F_{II}$ equal to ...
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Differential geometry and frenet formula

Given a curve C by its arclenght (vector $r(s)$), prove that $\frac{dT(s)}{ds} \times \frac{d^2 T(s)}{ds^2} = k^2 \omega$ where k is the curvature and $\omega$ is the darboux vector. I tried using ...
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Second fundamental form and normal surface curvature

Going through the following lecture notes from MIT on differential geometry - Link At equation 3.25, in order to calculate the magnitude of the normal curvature, an expression obtained from ...
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Example of “The eigenvalues of data covariance matrix, $\Phi^T\Phi$ measure the curvature of the likelihood function.”

I am reading PRML, Chapter 3.5.3, screen shot attached. I can understand the derivation and maths but hard to understand the meaning of "The eigenvalues of data co-variance, $\Phi^T\Phi$ matrix ...
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What kind of properties characterise “constant curvature” for a general geometry?

Question: What are some characteristic properties of constant curvature or flat spaces? Some motivation and context: There are a lot of things on manifolds that are called geometric structures, like ...
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Modify a curvature equation so that it could hold both positive and negative values)

I've got a question, it's about how we modify an equation so that the t inside it could take both positive and negative values So this is my curvature κ(t): (2t^(3)/(4+t^(4))^(3/2)I've attached the ...
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Electric field and curvature

My physic teacher said that In a conductor the electric field, which is non-zero only on the surface, is stronger where the curvature is bigger*. But he did not provide a mathematical proof for ...
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Equipping finite space with metric

I'm attempting to equip this finite space with a metric. The points in this space are defined as all the intersection points, (see image). I think the metric will be approximately the Euclidean metric ...
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Formula for the osculating conic of a plane curve

A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the ...
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Scale invariant curvature (plane curve)

Is there a form of curvature for a plane curve which is invariant under uniform scaling? Ideally, I am looking for a way to characterize the effective 'local eccentricity' of a plane curve so that [...
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Is there a term for Gaussian curvature based on geodesic curvature?

Gaussian curvature is usually defined as the product of maximal and minimal normal curvatures of curves through a point on a surface. What if we use geodesic curvatures of the ambient space instead? ...
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How Gaussian curvature is affected by a conformal map (using forms)

This is an exercise from Tu's book Differential Geometry. Let us say we have a two Riemannian manifolds $M$ and $M'$ of dimension 2 with a diffeomorphism $T:M\to M'$ between them. Say $T$ is conformal,...
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curvature and center of curvature of a quadratic parametric curve

A General Statement I read is: Given a parametric curve $$\vec r(s)=(x(s),y(s))$$ the tangent vector is represented by the first derivative, $$\vec r'(s)=(x'(s),y'(s))$$ Then the vector $\vec n=\...
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surface integral of mean curvature

I've problem with calculating integral mean curvature of an ellipsoid: $\iint_{S} \frac{(eg-f^2)}{EG-F^2} dS$ where S is surface of an ellipsoid. e,f,g and E,F,G are first and second fundamental ...
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Theorems involving upper Ricci curvature bounds

When scrolling through papers and books on Ricci-curvature I noticed that there is an incredible amount of theorems involving lower Ricci-curvature bounds and even a synthetic definition of a lower ...
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Is the Wikipedia formula for the Gauss equation correct?

I've been doing some computations and they don't seem to add up: Consider an $n$-sphere immersed in $\mathbb{R}^{n+1}$. Its Riemann tensor should be of the form $R_{abcd} = p *(g_{ac}g_{bd}-g_{ad}g_{...
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Scalar function from $XYZ$ coordinates

As part of the project, I have to calculate the Gaussian, mean, max and min principal curvatures. I understand that I need to use partial derivatives to get them and I understand how to do that but I ...
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Parabolic shot problem

In a parabolic shot thrown at an angle $\Theta$ with respect to the horizontal, it is known that the radius of curvature at the point where the object is thrown (labeled as point $A$) is $p_A$. ...
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Biconformal space and curvature

I've found very few contributions about the so called Biconformal Space, "a curved phase space". I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
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Curvature and torsion of circular helix

$c:r=r(s)$ $ c$ is circular helix iff $ det(r’’,r’’’,r^{(4)})= 0 $ i will prove one side but other side i said : Let $det(r’’,r’’’,r^{(4)}) = 0 $ then , $r^{(4)} = \alpha(s) r’’ + \beta(s) r’’...
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Does a surface $S$ diffeomorphic to a sphere have an umbilical point?

I should see if a surface which is diffeomorphic to a sphere in $\mathbb{R}^3$ always has an umbilical point. So, I start by locally parametrizing the sphere with a diffeomorphism $x: \Omega \to S^2$ ...
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In differential geometry, to what extent does the curvature tensor determine its associated connection?

In the absence of a metric, it is not clear to me to what extent does knowing the curvature tensor determine its associated connection? I would be satisfied knowing this for zero torsion. I'd like to ...
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Does existence of a parallel section on $E$ imply local existence of a parallel section on $\bigwedge^k E$?

Let $E$ be a smooth vector bundle equipped with an affine connection $\nabla$. Suppose that $(E,\nabla)$ admits a non-zero parallel section. I think that $(\bigwedge^k E,\bigwedge^k \nabla)$ does ...
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Curvature inequality involving a Curve within a disk

If a closed plane curve $C$ is contained inside a disk of radius $r$, prove that there exists a point $p \in C$ such that the curvature k of C at p satisfies $\lvert k\rvert \ge$ $1/r$. I understand ...
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Non-convex numerical optimization

I am looking for a way how to prove/show that one problem is easier to optimize(more robust to starting guesses) than another. Suppose I have two non-convex functions from $L_1,L_2:\mathbb{R_+}^8\...
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curvature tensor of involutive distributions

The Riemann curvature tensor is defined $$R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z.$$ Is it possible to simplify above expression under assumption that $X,Y\in D$, where $D$ ...
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Definition of Gauss curvature in the general setting

I'm currently reading about the 2nd fundamental form and sectional and Gauss curvature. The situation where all the sources I'm consulting agree is the following: Let $S$ be a 2-dimensional ...
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Mobius strip with constant negative curvature

Is there any simple model of the Mobius strip with a constant negative Gaussian curvature? There is an example on Wikipedia (https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Open_M%C3%B6bius_band), but ...
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Why circle of curvature is defined to be in the inside part of the circle?

We know that for a curve at a certain point,its circle of curvature is defined to be a circle whose curvature is same as the curve at that point and both the curve and the circle has same tangent at ...