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)
1,797
questions
0
votes
2
answers
36
views
Does any compact boundary become a minimal hypersurface iff each of the boundary component is a minimal hypersurface?
There are researchers who made the following assumptions in their paper on the Penrose inequality:
Let $M$ be a complete, connected Riemannian $3$-manifold and suppose that the boundary $\partial M$ ...
- 1,149
0
votes
0
answers
44
views
Proof that every analytic function describes a minimal surface
Is my proof corect?
Hypotesis: "Every analytic function describes a minimal surface."
Proof:
Every analytic function satisfies the Laplace equation:
$\nabla^2 u = \frac{\partial^2 u}{\...
1
vote
0
answers
40
views
Ants, cylinders and shape operator
For an oriented surface in $\mathbb{R}^3$ the curvature tensor can be expressed as
$$ R(X,Y)Z = \langle L(Y),Z \rangle L(X) - \langle L(X),Z \rangle L(Y) $$
where $X,Y,Z$ are vectors fields on the ...
- 494
1
vote
1
answer
38
views
Lorentzian manifolds of constant curvature
I know that Riemannian manifolds of constant curvature are locally isomorphic to the spherical, flat or hyperbolic space depending on the sign of the sectional curvature.
Reading Hawking and Ellis' ...
1
vote
1
answer
24
views
I don't understand why the curvature of a function with Hessian $H$ in a direction $p$ is $p^T H p$
I am reading a book that claims (without proof) that the curvature of a function $f(x)$ along a direction $p$ is obtained by projecting the product of the Hessian $H$ and $p$ onto the direction $p$ i....
- 1,229
2
votes
1
answer
95
views
Level-set formulation of inverse mean curvature flow (IMCF)
I'm thinking about an equivalent formulation of IMCF proposed by Gerhard Huisken and Tom Ilmanen in their 2001 work THE INVERSE MEAN CURVATURE FLOW AND THE RIEMANNIAN PENROSE INEQUALITY:
Let $M$ be a ...
- 1,149
2
votes
1
answer
37
views
Totally antisymmetric Riemann tensor
Is it possible for a Riemannian manifold to have a non-vanishing Riemann tensor that is totally antisymmetric in the four indices? Of course, the antisymmetry would imply that the Ricci tensor ...
- 135
0
votes
0
answers
50
views
Does Curved Edge exist for a smooth infinitely long right circular Cylinder?
This question is in the continuation of this question. As it is cleared from the comments of the respective question that an infinitely long cylinder which is also a right circular, is a smooth $3$D ...
- 343
0
votes
1
answer
47
views
Positive Gaussian Curvature
So my quastion is who does one show that if $M\subseteq \mathbb{R}^2$ is a manifold with positive gaussian curvature (G.C) at every point and if a curve $\gamma\subseteq M$ is such that $\gamma'\neq0$ ...
- 49
0
votes
0
answers
32
views
Constant Curvature Reparametrization
I am getting into differential geometry and came up with the problem which is unexpectedly difficult to solve. Suppose we have a curve $\alpha(t):[0,L]\rightarrow \mathbb{R}^2 $ parametrized by ...
2
votes
0
answers
82
views
Is this a correct way to derive the Riemann curvature tensor?
I wanted to see if I can use a simple idea and derive the Riemann curvature tensor by only using the covariant derivative, because I somehow got the impression that the derivations I read in several ...
- 331
2
votes
0
answers
49
views
Determinant of a Frenet curve
I have this problem:
Let $c$ be a Frenet curve in $\mathbb{R}^n$. Show that
$
\operatorname{Det}\left(c^{\prime}, c^{\prime \prime}, \ldots, c^{(n)}\right)=\prod_{i=1}^{n-1}\left(\kappa_i\right)^{n-i}...
- 489
1
vote
0
answers
22
views
Integrated Curvature on Manifolds in SageMath
Is it possible to integrate the Ricci curvature scalar on an arbitrary $n$-dimensional manifold in SageMath? Just straight up:
$$\int R dV$$
Assume that I know how to get the Scalar curvature a la ...
- 2,575
0
votes
0
answers
23
views
Finding maximum curvature of $Qe^{-ax}$?
Background
I have the following function: $H(x)=Qe^{-ax}$, where all parameters are positive.
Using Mathematica, I used the following formula to calculate $H(x)$'s curvature:
$H_{C}=\frac{|H''(x)|}{\...
- 279
7
votes
0
answers
147
views
If $S = P^{-1}(0)$ is a surface, show $\int |K|\leq 4\pi C(d)$
Let $P\in \mathbb{R}[X,Y,Z]$ be an irreducible polynomial, such that $S=P^{-1}(0)$ is a non compact regular surface. If $K$ denotes the Gaussian curvature, prove that:$$\int_S |K|\leq 4\pi C(d)$$ ...
- 168
0
votes
0
answers
36
views
Does a geodesic locally minimize the curvature?
Does a geodesic somehow minimize the local curvature? Lets say we have a smooth geodesic $\gamma$ with constant speed on a surface $S\subset\mathbb{R}^n$ and with $\gamma(t)=x$, then every other ...
- 1,154
0
votes
0
answers
9
views
Rewriting Laplacian powers $\Delta^l$ on a boundary using directional derivative along the normal
How would one go about rewriting $\Delta^l u$ on a boundary $\partial\Omega$ in terms of $\partial_n^{2l}$, where $n$ is the normal of $\partial\Omega$? I am asking since I saw in remark 2.11 of the ...
- 1,751
1
vote
1
answer
32
views
Verifying the Bakry-Émery criterion on spheres
Suppose that I am interested in a probability measure $\mu$ defined on the sphere $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$, whose density with respect to the normalised surface measure can be written ...
- 10.7k
1
vote
2
answers
61
views
Differences in mean curvatures for permuted coordinates of a manifold
Below is an example demonstrating the differences in mean curvatures and one other quantity, and below is my question. Please comment if you need me to explain in more detail or clarify notation, or ...
- 1,763
8
votes
1
answer
164
views
Reconciling different expressions for Riemann curvature tensor
[Note: This has been crossposted to Physics SE, but I haven't found a thourough explanation there so far, so I'm posted the question here as well] I'm using Einstein's summation convention throughout. ...
- 2,321
2
votes
1
answer
99
views
Which Riemann tensor should I choose?
Apparently there are two conventions for the (1,3) tensor called the Riemann curvature and they give opposite functions
$$T_p M \times T_p M \times T_p M \to T_p M :$$
$$ (x,y,z)\mapsto R(x,y)z.$$
...
- 760
1
vote
0
answers
42
views
Convexity and positivity of the second fundamental form
Let $\Omega$ be a smooth bounded open set in a complete Riemannian manifold $(M^n,g)$, $n\geq 2$, and denote by $N$ the outward unit normal for $\partial \Omega$.
Is it true that $\Omega$ is (strictly)...
- 4,964
1
vote
0
answers
30
views
Curvature of the graph of a function defined on a sphere
I am not an expert in differential geometry and was wondering: Given a function $h:\mathbb S^{n-1} \longrightarrow \mathbb R $ smooth, the graph of $h$ is an $(n-1)$-dimensional submanifold of $\...
- 71
0
votes
0
answers
33
views
Is there any natural way to turn flat space with torsion into curved space without it?
Say we have a smooth function $f: \mathbb{R}^n \rightarrow SO(n)$, whose value is asymptotically zero in all directions. In other words, we're specifying a rotation at every point of space, but ...
- 189
0
votes
0
answers
32
views
Why is curvature scale $O(1)$?
The question comes from book "NETLAB: Algorithms for Pattern Recognition". In section 2.3.1 on page 42-43, the book first introduces bracketing interval:
A bracketing interval is a triple $...
- 53
1
vote
0
answers
28
views
Question about ratio of Riemannian densities
I am studying the paper "On the MCP property of metric measure spaces" by S. Ohta (https://ems.press/content/serial-article-files/1508) and I have a question regarding a passage at page 813 ...
- 429
1
vote
1
answer
73
views
Can a spline be constructed given start and end points, normalized tangent vectors, and curvatures?
I want to create a smooth curve that starts at point $P_0 = (Px_0, Py_0)$ with normalized tangent $T_0 = (Tx_0, Ty_0)$ (where $\sqrt{Tx_0^2 + Ty_0^2} = 1$) and with curvature $κ_0$ (where positive ...
- 969
1
vote
0
answers
73
views
Problem with a Curvature Calculation on $S^3\rightarrow S^2$
Sorry this going to be mildly long, I am having trouble obtaining what I think I should when I pull back the curvature form $S^3$ to $S^2$ via a local section, and would like to see if someone can ...
- 1,425
1
vote
0
answers
64
views
Calculation of Riemann Curvature Tensor
I am reading the book: "Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers" (P.M.Gadea, J.Munoz Masque) Problem 6.5.1. Page 252-253:
Find the Riemann ...
- 69
1
vote
1
answer
43
views
Why does a path homotopy between distinct geodesics must pass through a long curve?
I'm working on problem from Do Carmo's "Riemannian Geometry" (Ex. 1, chapter 10). The problem, called Klingenberg's Lemma, puts a lower bound on the length of some curve in a homotopy, ...
- 512
3
votes
0
answers
79
views
Can we think of the Riemann curvature tensor as a tensor-valued 2-form and integrate it over a surface?
The usual physical interpretation of the Reimann tensor $R^\mu_{\ \ \nu \rho \sigma}$ at a given point $p$ on a manifold $M$ is that it inputs an infinitesimal vector $v^\nu$ and two other ...
- 5,738
1
vote
0
answers
45
views
Torsion on a flat connection (Geometric Intuition).
Start with the 2-sphere $\mathcal{S}^2$ with the standard $(\theta, \phi)$ chart ($\theta = 0$ is the North pole etc) and metric:
$$ds^2 = d\theta^2 + \sin^2 \theta d\phi^2$$
Remove the two poles from ...
- 991
1
vote
1
answer
35
views
How does curvature relate to angle measurement in hyperbolic geometry?
This question is about the relationship between curvature and angle measurement in hyperbolic geometry... Specifically, I am trying to understand the following excerpt from pp. 489-490 of Greenberg's ...
- 179
2
votes
0
answers
30
views
Prove that vector tangent to a curve always maintains the same angle, so the ratio between curvature and torsion is constant
I have a 3D curve parametrized by the arc length, $r(s)$, which I compute the curvature $\kappa$ and de torsion $\tau$ as:
\begin{equation}
\kappa(s) = ||\frac{dT}{ds}|| = ||r''(s)||\\
\tau (s)...
- 159
5
votes
1
answer
140
views
Calculating scalar curvature of a warped product $I \times N^n(k)$
I'm trying to work out the details of this example by R. Bryant. He considers a Riemannian manifold $(N ^n, h)$ of constant sectional curvature $k$ and constructs the following quadratic form on $\...
- 6,450
2
votes
1
answer
52
views
Curvature of Connection [exercise proofcheck]
I tried to solve the following question, but I get the feeling I am doing a few things wrong, but cannot really find out what.
So the exercise is:
Let $\alpha, \beta \in \Omega^{1}(M)$ for a manifold $...
- 53
0
votes
1
answer
69
views
Curvature defined as exterior covariant derivative of connection?
In Gauge Theory and Variational Principles by David Bleecker, the curvature of a $\mathfrak{g}$-valued 1-form connection is defined as $\Omega^\omega:=D^\omega\omega=(d\omega)^H$, where $\omega(d\...
- 277
2
votes
1
answer
50
views
Derive formula for conformal curvature
So I'm working through Needham's Visual Differential Geometry and Forms, and what is suggested as a simple exercise is to reduce the general Gaussian curvature formula
$$K = -\frac{1}{AB}\left(\...
- 335
1
vote
1
answer
105
views
Integration of Ricci curvature on a compact manifold.
$\mathbf {The \ Problem \ is}:$ Let, $(M,g)$ be a compact Riemannian manifold and $X\in \chi(M).$ Show that $\int Ric_g(X,X)\mu_g=\int ((tr(\nabla_.X))^2-tr(\nabla_.X\circ \nabla_.X))\mu_g.$
$\mathbf {...
- 2,037
3
votes
0
answers
29
views
Realization of the $2$-form as the curvature of some bundle.
Suppose $M$ be a manifold, and $\omega$ be any smooth $2$-form. My question is about the existence of the Lie group $G$ which satisfies that there is always a principal $G$-bundles $P$(or its ...
- 720
0
votes
0
answers
65
views
Chern class of a hypersurface in $\mathbb{C}P^3$
Let $X=\{[z_0,z_1,z_2,z_3]\ \big{|}\ [z_0,z_1,z_2,z_3]\in\mathbb{C}P^3,z_0^4+z_1^4+z_2^4+z_3^4=0\}$. $c_1(X)$ is the first Chern class of $X$. Prove that $c_1(X)=0$.
$\textbf{My try}$:
It's easy to ...
- 299
3
votes
1
answer
37
views
No Nash-Kuiper theorem for $\mathcal C^2$ isometries
Reading about the Nash-Kuiper theorem, I found the following statement in Y. Eliashberg and N. Mishachev's book Introduction to the $h$-Principle:
Is there a regular homotopy $f_t:S^2\rightarrow\...
- 2,513
2
votes
2
answers
82
views
Minimizing the maximum curvature: a real-life, liquid-filled pipe question [closed]
I've searched Stack Exchange for this question, but have only found Bezier curves which I don't think is what I'm looking for.
I have a real-world problem where I need to design a system of water-...
- 21
0
votes
0
answers
60
views
Radius of Curvature for Cam Profile
In the design of a radial design cam-follower mechanism, the radius of curvature of the cam profile is often used to identify areas of high contact stress.
The pitch circle of a cam is defined by ...
2
votes
0
answers
57
views
Ambiguity in trace of curvature 2-form matrix
My question is about computing the first Chern class from the trace of curvature 2-form matrix via the Chern–Weil theory. First Chern class is given by
$$ c_{1}=\left[{\frac {i}{2\pi }}\operatorname {...
- 21
1
vote
1
answer
40
views
Why non zero wedge term in curvature form?
The definition of the curvature form ( from Wikipedia ) is:
$$ \Omega = d \omega + \frac{1}{2} [ \omega \wedge \omega ] $$
I am puzzled about the rightmost term. According to what I have learned so ...
- 113
2
votes
0
answers
40
views
Gauss Equation in Coordinates
Consider a $(d+1)$-dimensional pseudo-Riemannian manifold $(\mathcal{M},g)$ and a Riemannian hypersurface $(\Sigma,h)$, i.e. $\Sigma$ is an embedded submanifold of $\mathcal{M}$ of codimension $1$ ...
- 1,992
4
votes
1
answer
103
views
Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces
Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
- 1,763
3
votes
1
answer
57
views
Expression of a metric tensor with constant curvature on surface
Consider the plane $\mathbb{R}^2$ together with a metric $\mathsf{g}$. Choosing suitable local coordinates $(x_1,x_2)$, $\mathsf{g}$ takes the form :
$$
\mathsf{g}=r(x_1,x_2)\left((dx_1)^2+\epsilon (...
- 344
2
votes
1
answer
84
views
Compact surface with Gaussian curvature 0 everywhere
I'm trying to give an example of a compact surface $S\subset\mathbb{R}^3$ with constant zero Gaussian curvature for all $p\in S$. I know that a plane would have constant zero Gaussian curvature, but ...
