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)
1,288
questions
0
votes
0answers
28 views
3 definitions of curvature, which is it?
On the youtube lectures on differential equations by Claudi Arezzo the curvature is defined as:
(He assumes the curve is arc length parametrized maybe this matters)
$k(s) = |\alpha''(t)|$
On the ...
2
votes
1answer
41 views
Proving $R_{ij;m}=g^{kl} R_{ikjl;m}$.
In the coordinate $\{x^i\}$, the Riemann curvature tensor can be written as
$$
R=R_{ijkl}\,dx^i\otimes dx^j\otimes dx^k \otimes dx^l
$$
and the Ricci curvature can be written as
$$\text{Ric}=R_{...
-2
votes
1answer
33 views
How do I determine the curvature of an arc length parameterized curve in the $xy$-plane? [closed]
I have a 2D curve in the $xy$-plane, which was arc length parameterized numerically, and fitted by cubic splines for both $x$ and $y$.
If one of the segments of the cubic spline is:
\begin{align}
x&...
1
vote
0answers
38 views
Curvature plane curve after applying linear map
Let $\alpha$ be a plane curve, whose curvature can be determined using the following formula:
$k_{\alpha}(t) = \frac{det(\alpha^{''}(t),\alpha^{''}(t))}{|\alpha^{'}(t)|^3}$
Let $L$ be a linear map ...
-2
votes
1answer
34 views
Compute the coefficients of 1'st and 2'nd fundamental form of the surface.
Let $x(u,v)= (\cos u.\sin v$, $\sin u .\sin v$, $\cos v$), $\;0 < u < 2\pi$ , $0 < v < \pi$ be a parametrized surface S.
1-Compute the coefficients of the first fundamental form(I) of ...
0
votes
0answers
27 views
Meaning of Gauss curvature of a surface in general 3-manifold
Let $(\Sigma,g)$ be a surface isometrically embedded in a $3$-dimensional Riemannian manifold $(M,\bar{g})$. I'm particularly interested in the case where the ambient space $(M,\bar{g})$ is not the ...
0
votes
1answer
42 views
Curvature and torsion of coordinate curve on the sphere
find the curvature and torsion of a $v=v_0$ (= constant) coordinate curve on the sphere
$x(u,v)= (a.\cos u.\sin v$, $a.\sin u .\sin v $, $a.\cos v$), $\;0 < u < 2\pi$ , $0 < v < \pi$
I ...
0
votes
0answers
13 views
tanget bundle with sasaki metric is Kahlet iif M is locally flat [closed]
I'm having a hard time proving the following
If $M$ is an n-dimensional indefinite Riemannian manifold whose metric g has index s, then the metric of Sasaki $g^{D}$ is an indefinite metric on $TM$ ...
0
votes
0answers
12 views
How to derive the expressions of $u_\phi$ and $u_t$ for a Kerr metric?
The Kerr metric in and near the equatorial plane can be expressed as (Novikov and Thorne (1973))
$$ds^2=-\frac{r^2\Delta}{A}dt^2+\frac{A}{r^2}(d\phi-\omega dt)^2+\frac{r^2}{\Delta}dr^2+dz^2$$
I am ...
1
vote
0answers
26 views
LaplaceāBeltrami operator and Ricci curvature
I am taking a course on General Relativity which is more mathematically oriented than physically, and in one of the lecture notes (discussing Riemann tensor, curvature and all that) this identity is ...
0
votes
0answers
8 views
proof that holomorphic curvture is constant if the curvature tensor R is null
I am trying to prove that holomorphic curvture is constant if and only if curvature tensor R is null
I already proved one implication and I can't prove the other
can you halp please?
0
votes
1answer
61 views
Is total curvature of a closed space curve a multiple of $2\pi$?
For a regular closed space curve $x:[0,L] \rightarrow \mathbb{R}^3$ parametrized by arc-length $s$, we define total curvature
$$
K = \int_0^L \kappa(s) ds
$$
In particular, when $x$ is a plane ...
0
votes
0answers
19 views
Simplification of the derivative of the mean curvature
I'm using the RK method to calculate the thin film profile. there is a complex interface curvature term, as shown in 1, Īŗ. The final formula is a fourth-order ordinary differential equation which ...
0
votes
0answers
8 views
Is there a way to estimate the shape or curvature of a function given some data points?
I have a series of measurements (data points), we can call them Ys for now, and two input variables m and n, such that f(m,n) = Y. Is there an easy way to compute the shape or curvature for function f ...
0
votes
0answers
21 views
Computing principal curvatures from local surface normals
Given a surface in 3-space described solely by a 2D (UV) grid of surface normals, I want to compute the principal curvatures (and optionally their directions in UV-space) for each point on the grid. ...
4
votes
1answer
45 views
From Riemann tensor to Ricci tensor and vice versa
Is it possible to find the Riemann tensor using the Ricci tensor and also to switch from Riemann $(0,4)$ to Riemann $(1,3)$?
I am not clear if these operations can only be carried out in one sense, I ...
0
votes
0answers
34 views
Variation of the (Chern) curvature with respect to the metric
Let $E\rightarrow X$ be a holomorphic vector bundle, for any Hermitian metric $h$ on $E$ we denote by $F_h$ the curvature of the Chern connection associated to $h$. Fix a metric $h_0$ and consider a ...
0
votes
0answers
46 views
Sectional curvature of the sphere via geodesic variations.
I'm working with the sphere, $S^2$, with the induced Euclidean metric from $\mathbb{R}^3$. I have that $u,v \in T_p(S^2)$, for some $p \in S^2$, with $u$ and $v$ orthonormal. I have shown that
$$F(s,t)...
0
votes
0answers
17 views
Found curvature of a curve
I have a series of points that define a curve, and I want to find the radius of curvature in each point. I thought to calculate the spline interpolation, and so use it to calculate the curvature in ...
0
votes
0answers
12 views
Calculating average curvature for a set of points
I am starting with a set of $(x,y)$ points, example here.
I would like to get a value for the average curvature of the set of points. My strategy was to find values for the parameters ($h, k, r$) in ...
2
votes
0answers
55 views
Relationship between a certain scalar and any known notions of curvature in geometry
Disclaimer. I'm not an expert geometer, so feel free to fix my language if I use the wrong words...
Let $S$ be a smooth $(n-1)$-dimensional surface in $\mathbb R^n$, with "inside" $A \subseteq \...
0
votes
0answers
19 views
Evaluating the double integral of a cross product (to prove Gaussian Curvature)
My end goal is to show why Gauss's original equation for curvature is equivalent to the modern definition. Stated differently, explain why $$K= \lim_{A\to p}\frac{Area(n(u))}{Area(U)}=\frac{eg-f^2}{EG-...
0
votes
0answers
18 views
Relationship between area elements and infinitesimal rotations - and meaning of the symmetry of the curvature tensor
I am going to use index notation, since its more descriptive in this case.
If we have a (pseudo-) Riemannian manifold with metric $g$, the curvature tensor of its Levi-Civita connection satisfies the ...
0
votes
1answer
34 views
Obtain the parametric equation of the curve with curvature proportional to length.
Task: Obtain the parametric equation of the curve, the curvature of which varies in direct proportion to the length of the arc.
Please help me decide.
1
vote
1answer
44 views
Mean curvature of a surface - Why does it equal this formula?
The mean curvature of a surface is defined as $$H = \frac{1}{2}(k_1+k_2)$$ where $k_1$, $k_2 $ are the principal curvatures.
More abstractly, it is the trace of the shape operator $$H = tr(S)=\frac{...
0
votes
0answers
19 views
Covariant Derivative vs Partial Derivative Contradiction
I am doing gradient descent with respect to se loss function L. I have found that the partial derivative wrt some of the parameters is zero whilst the covariant derivative wrt the same parameters is ...
0
votes
0answers
22 views
Approximating curvature of a Riemann Manifold
I have read that given a 2-manifold $M$, it is possible to approximate the Gaussian Curvature at any point $p$ in $M$ using simplicial complexes made of triangles in such a way that as the maximum ...
0
votes
0answers
21 views
Toponogov's Theorem
The following picture comes from the book "Ricci Flow and the Sphere Theorem". Here, the author mentioned the Toponogov's theorem.
Here is the content of the reference.
How can this imply the ...
0
votes
1answer
27 views
finding the principal curvature
I am trying to find the principal curvature and its direction of:
$$x^2+y^2-z^2=0$$ I tried to re-parametrize with $r(\theta,z)=(z^2\cos(\theta),z^2\sin(\theta),z^2)$
But the second fundamental ...
0
votes
0answers
30 views
How to find rotation matrix of a plane curve of varying curvature?
I'm able to find a rotation matrix with respect to a fixed basis for a plane curve of constant curvature (example - circle) or a straight line (zero curvature).
But in the case of a sinusoid, ...
1
vote
0answers
39 views
How would we perceive space from within the hypersphere $S^3$?
Imagine being in a relatively small 3-sphere. Since geodesics behave differently from euclidean space, light will follow different trajectories, and I am wondering what the impact would be on our ...
0
votes
0answers
12 views
Moving from Paramertic Space to other space
I have a scalar field F=f(t,s) in a parametric space (tā[0,1], sā[0,1]). I am interested in calculating the normal and curvature at a point (t1,s1). This, I do by $\hat{n}=\frac{F_x\hat{i}+F_y\hat{j}}{...
1
vote
1answer
14 views
How to show composition of Sigmoid function and $\log x + \alpha$ is concave?
Let $\textbf{Sigmoid}(x) = \sigma(x)=\frac{1}{1+e^{-x}}$. How can we show $\textbf{Sigmoid}(y)$ is concave where $y=\log x + \alpha$ and $\alpha\geq 0$?
My try:
We know $\sigma'(x)=\sigma(x)(1-\...
0
votes
0answers
78 views
Is there a way to visualize the geometrical meaning of Riemann curvature?
Riemann curvature depends on position, on metric, and on a plane (replacing the two linear independent vector used in calculating it with any not degenerate linear combination of themselves we got the ...
1
vote
1answer
39 views
Trouble finding osculating circle
I'm presented with finding the equation of the osculating circle at the local minimum of $\mathbf f(x) = 3x^3-9x^2+5x-1 $.
Finding the local minimum wasn't that hard; I take the first derivative of $\...
1
vote
0answers
11 views
Constant sectional curvature of $I \times_f S^n(1)$
Let $M = I \times_f S^n(1)$ be the warped product Riemannian manifold, where $I$ is an interval and $S^n(1)$ the $n$-dimensional unit sphere. I have to find a sufficient and necessary condition on $f$ ...
2
votes
1answer
23 views
Can't find where my error is: Find $k(t)$ given $\mathbf r(t) = \langle3t^{-1}, 6, t\rangle$.
I began with the formula: $k(t) = \frac{||\mathbf r'(t) \times\mathbf r''(t)|| }{ ||\mathbf r'(t)||^3}$.
My $\mathbf r'(t) = \langle -3t^{-2}, 0, 1\rangle$ and $||\mathbf r'(t)|| = \sqrt{9t^{-4}+1}$.
...
1
vote
1answer
43 views
Find out the maximum principal curvature of parametric surface: $P(u,v)$
A parametric surface is defined as
$$X=140u+20v-40uv-20, \ \ \ Y=80-80v \ \ \ \ Z=50-10u-50v+10uv$$
Where, $0\le u,v\le1$
Find out the maximum principal curvature of given surface.
My Try:
...
1
vote
0answers
46 views
Is the maximum sectional curvature smooth?
Given a smooth Riemannian manifold (M,g), is the function $f:M \rightarrow \mathbb{R}$, defined as the maximum of sectional curvatures at each point $m \in M$, smooth?
I think for each point $m \in ...
0
votes
2answers
61 views
Is there a spiral whose arc-length between two points on it is proportional to the difference in its radius of curvature between these two points?
This is a strange spiral, which is from the differential equations as follows
$$
\left\{\begin{aligned}
\dot x-\dot y=\ddot x\sqrt{\dot x^2+\dot y^2} \\
\dot x+\dot y=\ddot y\sqrt{\dot x^2+\dot y^2}
\...
0
votes
1answer
36 views
Index gymnastics with the Riemann tensor
We would like to obtain the Ricci tensor from the Riemann tensor.
In most books are contracted the first index with the third one, the second index with the fourth one. Following the same convention, ...
0
votes
1answer
28 views
Usage of the symmetry of the Riemann tensor
On the way to derive the energy-momentum pseudotensor the expression $(154)$ must be derived from $(153)$. $U^{ijkm}$ has the same symmetry properties as the Riemann tensor $R^{ijkm}$. What symmetry ...
1
vote
0answers
22 views
Symplifying the expression using the symmetry
$U^{ijkm}$ has the same symmetry properties as the Riemann tensor $R^{ijkm}$.
Let me remind the symmetries of the Riemann tensor, even most of you know it. As I know most common symmetry properties ...
0
votes
0answers
15 views
Does the expression of the second fundamental form of a regular surface in 3D involve the first fundamental form?
I am confused about the second fundamental form of a regular surface in 3D.
The first expression I saw (here) was that for any tangent vector $\mathbf{w}$:
$$
\mathbf{II}(\mathbf{w}) = \mathbf{w}^T \...
2
votes
1answer
47 views
For any Riemann Curvature $K$, does there exists an open set $G$ such that every point $p$ in $G$ has Curvature $K$?
For $2$-Dimensional manifolds, for any curvature $K$, we can create an open ball $G$ such that every point has the same curvature $K$, giving rise to spaces that are spherical, flat, or hyperbolic. ...
2
votes
2answers
69 views
Curvature and turning number of $t \mapsto (\cos(t), \sin(3t))$
Is the closed curve with period $2 \pi$
$$
\delta(t) := (\cos(t), \sin(3t))
$$
regularly homotopic to the positively traversed unit circle $(\cos(t), \sin(t))$?
From this question I now know that ...
1
vote
1answer
26 views
Find the curvature of the ellipse by the explicit equation $y=\pm\frac{b}{a}\sqrt{a^2-t^2}$
Find the curvature of the ellipse
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
at the point $P=(a,0)$.
Let $x=t$ and $y=f(t)=\pm\frac{b}{a}\sqrt{a^2-t^2}$
I want to find the curvature, $\kappa$, of the ...
0
votes
0answers
10 views
Would a random triangular mesh always, approximate to a platonic solid?
I was thinking about if a random triangular mesh could approximate the curvature of 2D manifold. If we assumed every link was 1 unit long. (This mesh is not embedded in some space, it is an abstract ...
4
votes
1answer
55 views
Are there geodesic triangles on surfaces with non-constant curvature with angle sum 180?
I've been reading about differential geometry and the Gauss-Bonnet theorem to write a paper for my geometry class and am interested specifically in geodesic triangles on surfaces.
I was wondering if ...
3
votes
1answer
62 views
Curvature Formula Proof By Definition
Question: Use Definition 3.2 to prove Theorem 3.4.
Definition 3.2 āThe signed curvature $k(s)$ of a plane curve $ \alpha: I \rightarrow \mathbb{R^2}, \alpha(u)=(x(u),y(u))$ is defined by $tā(s)=k(s)...
