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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Weyl tensor on isotropic points
Is it true that the Weyl tensor vanishes on isotropic points?
In a riemannian manifold $(M,g)$, by Weyl tensor $W$ I mean $W=R-P \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} g$, where $R$ is the ...
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Verify the equation $BO/MN= (NH/LH)(KL/MN)$ in relation to curvature?
Verify that $BO/MN= (NH/LH)(KL/MN)$, and state why
$$ \frac{BG}{GM}= \frac{HN}{HL} \frac{KL}{MN}$$
I am unsure how to even start this problem. Any help would be appreciated.
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How do you solve an equation in relation to curvature?
Image of the the points and lines. Show that
$$ \frac{MN}{KL} = 1 + \Big( \frac{LN - KM}{KL} \Big). $$
I am unsure how to even start this problem. Any help would be appreciated.
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Word for curvature change along a path?
In natural coordinates, the curvature $k$ is defined as the along path rate of change of the unit tangent vector, or equivalently the along path change of angle between the x-axis and the velocity ...
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1answer
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Definition of curvature and intuition
I am trying to derive the formula for curvature by finding the radius of best fitting circle. So let $$ \vec{r} \left(t \right)$$
be the parametrization of the curve in question. Let's define tangent ...
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Surfaces that maintain constant mean curvature under uniform outward flow
Over on Physics.SE there is an interesting question about electrostatic configurations where all electric field lines are straight. Clearly, setups with spherical, cylindrical, or planar symmetry are ...
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Mean curvature of a surface same as inverse diameter of equivalent sphere?
Let $A$ denote, in the plane, the area enclosed by the perimeter $P=\partial A$, and $K=\frac{1}{2\pi} \int_0^{2\pi}\kappa(\theta)d\theta$ the mean curvature of $P$ (i.e. the average of the local ...
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1answer
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About the definition of vertex
Following is the definition of a vertex from $\textit{Elementary Differential Geometry}$ by Pressley
Definition.A vertex of a curve $\gamma(t)$ in $\mathbb{R^2}$ is a point where its signed curvature ...
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Question about local parametrization near a point
So I have given a set $M=\{(x,y,z)\in\mathbb{R^3}:\frac{x^2}{R^2}+\frac{y^2}{r^2}+\frac{z^2}{r^2}=1\}$ where $0<r<R$. and I have to find the Gaussian curvature, which we do using a formula with ...
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Submanifold in $R^n$ with positive sectional curvature
I’m trying to prove that for a compact hypersurface $M⊂R^n$($n>2$),there exists a point p where $sec(σ)>0$ for all 2-planes $σ ⊂T_pM$.
My idea is :
Since M is compact, we can choose a maximal ...
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1answer
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Shifrin Differential Geometry Exercise $1.2.27$ — A Differential Equation For Bikes
The Question
Suppose the front wheel of a bicycle follows the arclength-parametrized plane curve $\vec{\alpha}$. Determine the path $\vec{\beta}$ of the rear wheel, $1$ unit away. As the hint explains,...
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Nonpositive curvature with closing geodesics.
Let $M$ be a complete Riemannian manifold, $x \in M$ a point, and
$\gamma_s(t)$ be a family of geodesics starting at $x$ at time $t=0$,
$s \in [-\epsilon,\epsilon]$. The Jacobi field $J(s,t)$ of this ...
2
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1answer
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What is the point of giving a tensor identity in normal coordinates?
I have been confused about this for some time. For example, the curvature tensor in general for the Levi-Civita connection of $g$ in the local frame $\{\partial_i\}$ induced by the coordinates $\{x^i\}...
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Generalizing Stewart's theorem to timelike triangles in Lorentzian spaces of constant curvature
Stewart's theorem describes the length relations of the sides of a triangle and any one of its cevians in flat space ($n$-dimensional Euclidean space), and also in flat spacetime ($1 + n$-dimensional ...
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1answer
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Diffeomorphic manifolds with the same constant curvature but not isometric
I am looking for a two Riemannian manifolds which are diffeomorphic, have the same constant curvature but they are not isometric.
I thought about the product manifold $(-\frac{\pi}{2}, \frac{\pi}{2})^...
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Ricci curvature tensor arguments
i do not understand the application of the Weizenböck identity,
$$
\Delta = \nabla^*\nabla + \text{Ric},
$$
where $\Delta = \text{d}\delta+\delta\text{d}$. It is applied like this
$$
\eqalign{
(\text{...
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Is the gaussian curvature always the product of it's principal curvatures?
I've found 2 definitions for the gaussian curvature. That being:
...
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1answer
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Non-compact manifolds with everywhere positive curvature
What are examples of non-compact complete Riemannian manifolds with everywhere positive curvature?
Can you give examples of 2-dimensional surfaces in $\mathbb{R}^3$ with this property?
Note that by ...
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What is the meaning of Christoffel symbols in the context of manifolds?
I am trying to re-learn differential geometry and I ran into a conceptual wall regarding Christoffel symbols. I think I understand their meaning in the context of a change of coordinates, but I fail ...
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1answer
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Does diffeomorphism change sectional curvature too much?
Pertaining to sectional curvature:
$f: M \to M$ is a $C^1$-diffeomorphic map on Riemannian manifold $M$, $K_p(u,v)$ is the sectional curvvature at $p \in M$ and $u,v \in T_pM$ independent. Want to see ...
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Geometric interpretation of the Riemann tensor with all lower indices
The only interpretation of the Riemann tensor I know of is that
${R^a}_{bcd}$
is the $a$ component of the deviation of the $d$ vector after transport around a $bc$ loop. I guess that's the intuition ...
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1answer
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Prove the curve is a part of circle [duplicate]
Given a curve $y$ having constant curvature with $y=f(x)$ with $f''(x) > 0 $. I have to prove that this curve is a part of a circle.
My Try
Assume curvature of the curve as $c$. Then, I can write $$...
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1answer
100 views
Expressing third covariant derivatives in terms of second covariant derivatives
I'm following the tutorial at this link, where the author states:
These follow from the various way one can iterate covariant derivative
$$\nabla^3_{xyz}s = \nabla^2_{xy}(\nabla_zs) - \nabla_{\nabla^...
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Geometric intuition for Hessian matrices
I have been learning about multivariate calculus on my own. I have learned, notation-wise the properties of concave and quasi-concave functions. But I am finding it very difficult to find geometric ...
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Can I simply replace the first and second derivatives with the gradient and laplacian in the curvature of the graph of a function?
Can I simply replace the first and second derivatives with the gradient and laplacian in the curvature of the graph of a function to generalize to a multivariable curvature functional of the graph of ...
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2answers
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Where does the 3rd power of the denominator of the curvature of the graph of a function come frome?
According to Wikipedia, the curvature of the graph of a function $f$ is given by the following ratio (assuming second-differentiability).
$$\mathscr{k}_f(x) = \frac{f^{\prime\prime}(x)}{(1 + [f^{\...
1
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1answer
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Variational Formulation of Yamabe Problem Equivalence.
Suppose we have an n-manifold ($M$, $g_0$) with scalar curvature $S_0$. I'm aware that the Yamabe problem can be classed as the problem of minimising the following Einstein-Hilbert Functional over the ...
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Curvature formula for general regular curve
If we have a curve $\gamma :(a,b)\rightarrow \mathbb{R}^n $. Then we have the same curve but with an arc length parametrisation $\hat{\gamma} $ such that $\gamma =\hat{\gamma } \circ s .$ (So $\hat{\...
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Number of dimensions to fully embed a (possibly pseudo-) Riemannian manifold reflected in intrinsic quantities?
Say we're given a (possibly Pseudo/Semi-) Riemannian manifold $M$. There are two (equivalent) ways to go about analyzing it. The “old” way is consider it as an embedding withing a higher dimensional (...
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1answer
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Why does a helix have a radius of curvature of $(r^2+h^2)/r$?
I can follow the math to calculate the radius of curvature of a helix, but what has me confused is there is no $z-$component to the normal vector, so the normal vector points at the $z$ axis at all ...
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Curve on a sphere determined by geodesic curvature
Is it possible, given an arc-length parametrized curve $\gamma : [0, 1]\to S^2$, and given $\gamma(0), \gamma'(0)$, to show that $\gamma$ depends only on its signed geodesic curvature, and if so is ...
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Uniqueness (up to rotation) related to the signed geodesic curvature
Take $S^2$ to be the unit sphere oriented by its outward normal. Let's consider two
unit-speed parametrized curves $\alpha=\alpha(s)$ and $\beta=\beta(s)$ in $S^2$. If they have the same signed ...
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1answer
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Finding the curvature & torsion of the derivative of a smooth regular curve in $\mathbb{R}^3$
Let $\beta(s)$ be a smooth regular curve in $\mathbb{R}^3$ parameterized by arclength with nowhere vanishing curvature. Let $\gamma(s) = \beta'(s)$. Find $\kappa_\gamma(s)$ and $\tau_\gamma(s)$ in ...
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Orthogonal decomposition of the connection/covariant derivative with respect to normal vector fields
Let $M$ be a smooth manifold, $N$ be a Riemannian manifold and $\iota: M \to N$ be an immersion. Let $g$ be the metric on $N$ and $\nabla$ be the Levi-Civita connection with respect to $g$. It is a ...
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Proving that $\frac{\nabla^2 u}{ds \, dt} - \frac{\nabla^2 u}{ds \, dt} = k_{\nabla}(\frac{d\gamma}{ds}, \frac{d\gamma}{dt})(u)$
I am currently trying to make Exercise 20 of these lecture notes.
It should be a very simple exercise, but I am swamped due to the amount of bookkeeping. I fear I am missing a very simple nuance that ...
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2answers
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Find the curvature if $x= \cos t$ and $y=\ln 2t$ at $t=\pi$
Find the curvature of the following curve:
$$ \begin{cases} x= \cos t \\ y=\ln(2t) \end{cases}$$
at $t=\pi$.
Can anyone help me with how to solve the question? Because I am confused if I need to put ...
1
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1answer
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Proving that the curvature of the induced connection satisfies $k_{\nabla^{*}}(X,Y) = k_{\nabla}(X, Y)^{*}$.
I am trying to make Exercise 19 of these lecture notes.
I'll briefly summarize the question below.
Exercise Let $V$ be a dinite dimensional vector space and $A \in \mathrm{End}(V)$. We consider
$$
A^*...
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1answer
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The curvature of a tangent to a curve considered as a curve in its own right
Suppose we have a curve $c:I \rightarrow \mathbb{R^3}$ that is parametrised with respect to arc length. We then consider $T,B,N$ (tangent, binormal and normal) as curve in there own right each defined ...
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A curve with all points umbilical and binormal vector in the tangent plane must be planar
Suppose you have some surface $S$ with a curve $c$ in it that allows for a Frenet triad. If we know that for every point along the curve $c(t)$ that $T(t) \in T_pS$ (always the case) but also that $B(...
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Mean Gaussian Curvature using non-unit vectors.
Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak.
Let us suppose that $\lambda_{h|}^i$
is not a unit vector and therefore, the mean curvature $M_h$ in
this ...
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1answer
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Constant curvature and parallel curvature tensor
Let $(M,g)$ be a pseudo-Riemannian manifold. We say that this is a space of constant curvature if the sectional curvature $K(p,S)=\frac{\langle R(X,Y)X,Y\rangle}{X^2Y^2-\langle X,Y\rangle^2}$ is ...
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Prove that there is a closed geodesic in the “inside” of a torus
I want to show that there exists a closed geodesic on an arbitrarily chosen surface where the Gaussian curvature is negative (on the entire surface).
One example would be the "inside" of a ...
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1answer
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sum of covariant derivatives implies constant Ricci curvature
Let $R_{ij}$ denote the Ricci tensor . If
$R_{ij,k}+R_{jk,i}+R_{ki,j}=0$ , prove that the scalar curvature $R$
is constant .
A straightforward calculation gives $$R_{ij,k}=\frac{\partial R_{ij}}{\...
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Given area and volume, find constants in CMC ode
EDIT1:
An axi-symmetric surface meridian encloses ( by rotation) maximum volume $V$ for given surface area $A$.
It is known that the principal curvatures of a constant mean curvature CMC (DeLaunay) ...
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1answer
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Characterization of disks of constant curvature and whose boundaries have constant geodesic curvature
Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose boundary has constant geodesic curvature, then $D$ is isometric to some geodesic ball of the ...
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1answer
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Two Definitions of the Weyl Tensor
I'm reading "Textbook in Tensor Calculus and Differential Geometry" by Prasun Kumar Nayak and came across the Weyl tensor/projective curvature tensor $C_{kijl}$. The book states that
$$C_{...
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Existence of point with high curvature.
When working on an exercise, I stumbled upon the following "intuitive fact" which I couldn't prove:
Let $\alpha, \beta:[a, b] \to \mathbb{R}^{2}$ be two smooth curves with $\alpha(s) = \...
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1answer
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Discrepancies in defining the curvature of a Cartesian function
I was thinking about how one might be able to define how curved a function was. I thought of two ways:
Method 1
The angle of a tangent of a function is $\arctan\left(\frac{dy}{dx}\right)$ (which we'...
2
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1answer
46 views
Mean curvature of graph over its tangent plane
Let $S$ be a regular surface in $\mathbb{R}^3$ and $p\in S$ a point on the surface.
By the implicit function theorem $S$ can be locally written as a graph of a function, e.g. $V\cap S = \{ (x,y,z) \in ...
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Parallel translation of a vector on a surface: parallels of the sphere
In V.I. Arnold's Mathematical Methods of Classical Mechanics the first appendix is on Riemannian curvature. He starts with parallel translation of a tangent vector to a surface:
Along a geodesic the ...
