## 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 case of torsion, namely that torsion is the covariant differential of a differential form. To me that does not make sense, because $$\Omega=d\omega+\frac{1}{2}[\omega,\omega]$$ and there is no representation in which that $$\frac{1}{2}$$ can arise.

Any explanations?

## Calculating points of a curve given by its curvature [on hold]

I try to draw a curve using it’s curvature. The curvature is given in a function: f(x), f(0) meaning the curvature at the start of the curve. This (linear) function describes the curve from start to finish and is given as: curvStart to curvEnd. Negative curvature = turn to the right, positive curvature = turn to the left.

I also have the starting coordinates for a curve (x and y). What I want to do now is getting points that I can plot on a 2D plane from these informations.

For example: Curvature from -0.5 to 0, starting coordinates (0,0) and I want 3 points would result in a point at the start (0,0), a point at the end of the curve and one exactly in the middle of the curve.

I hope you can understand what I want to do and I am very sorry for my bad explanation, I am lacking the terms and the understanding of the topic to describe it properly. Sadly most “tutorials” online only care about the curve to curvature part.

## Directional Curvature

What is Directional Curvature and how can I achieve it for any function? A common approach with an example would be much appreciated.

(Reference: I am reading “The Non-convex Geometry of Low-rank Matrix Optimization” paper and in section 1.2, Weighted PCA part, I got stuck. Link to the paper: https://academic.oup.com/imaiai/article/8/1/51/4951409)

## Explicit computation of connection & curvature matrix

I have recently learned the generalized Gauss-Bonnet theorem, which states that:

$$$$\int_M \text{Pf}(\Omega) = (2\pi)^n\chi(M),$$$$ where $$n$$ is half the dimension of an even dimensional, compact, Riemannian manifold.

Here, $$\Omega$$ is the curvature matrix of 2-forms determined by the Riemannian metric $$g$$ and some metric compatible connection $$\nabla$$, and $$\text{Pf}(\Omega)$$ is the Pfaffian.

By Chern-Weil, we know that our choice of $$\nabla$$ does not make any difference.

Question: The above integral, if the dimension of the manifold in question is 2, better reduce to the Gauss-Bonnet theorem that we know and love: $$$$\int_M KdA = 2\pi\chi(M),$$$$ where $$K$$ is the Gaussian curvature. But I am not sure how I can carry out the computation necessary to get there…

More specifically, I know that if the dimension is 2, then $$\text{Pf}(\Omega)$$ is gonna be a 2 form, more precisely, some number times $$\Omega_1^2$$, the upper-right entry of $$2\times 2$$ curvature matrix. If all were to work, this 2-form better be the form $$KdA$$.

By Chern-Weil, we may as well assume that the connection in question is Levi-Civita. Then the Theorema Egregium allows us to write $$K$$ in terms of $$g$$ and the associated Christoffel symbols.

My problem is that I don’t know how to carry out this explicit computation… Could you help me with this?

## When are principal lines of curvature geodesics?

Let $$S$$ be a smooth surface embedded in $$\mathbb{R}^3$$. When are (some of) the principal lines of curvature geodesics on $$S$$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal line, is a geodesic? And perhaps the (red) min principal line connecting the two umbilical points is a geodesic?

Image from Jorge Sotomayor.1

Is there any $$S$$ all of whose principal lines of curvature are geodesics?

1Sotomayor, Jorge. “Historical Comments on Monge’s Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in $${\mathbb R}^ 3$$.” arXiv Abstract (2004). São Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143.

## Lemma $3.2$ – Mean curvature flow singularities for mean convex surfaces

This is a lemma of the paper “Mean curvature flow singularities for mean convex surfaces” by Gerhard Huisken and Carlo Sinestrari (the paper is available here):

$$\textbf{Lemma 3.2.}$$ Suppose $$(1 + \eta) H^2 \leq |A|^2 \leq c_0 H^2$$ for some $$\eta, c_0 > 0$$ at some point of $$\mathscr{M}_t$$, then we also have

(i) $$-2Z \geq \eta H^2|A|^2$$;

(ii) $$|H \nabla_i h_{kl} – \nabla_i H h_{kl}|^2 \geq \frac{\eta^2}{4n(n-1)^2c_0} H^2 |\nabla H^2|$$

My doubt is concerning to item $$(ii)$$ and below is the argument given by the authors

We have (see [10, Lemma $$2.3$$ (ii)], reference [10] is available here)

$$|H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 \geq \frac{1}{4} |\nabla_i H \ h_{kl} – \nabla_k H \ h_{il}|^2 = \frac{1}{2} (|A|^2 |\nabla H|^2 – |\nabla^i H h_{il}|^2).$$

Let us denote with $$\lambda_1, \cdots, \lambda_n$$ the eigenvalues of $$A$$ in such a way that $$\lambda_n$$ is an eigenvalue with the largest modulus. Then we have $$|\nabla^i H \ h_{il}|^2 \leq \lambda_n^2 |\nabla H|^2$$ and

\begin{align*} |H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 &\geq \frac{1}{2} \sum\limits_{i=1}^{n-1} \lambda_i^2 |\nabla H|^2 = \sum\limits_{i=1}^{n-1} \lambda_i^2 \lambda_n^2 \frac{|\nabla H|^2}{2\lambda_n^2}\ &\geq \sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \frac{|\nabla H|^2}{2(n-1)|A|^2}\ &\geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2 \frac{|\nabla H|^2}{n(n-1)|A|^2}\ &= \frac{(|A|^2 – H^2)^2}{4n(n-1)|A|^2} |\nabla H|^2 \geq \frac{\eta^2 H^2}{4n(n-1)c_0} |\nabla H|^2. \square \end{align*}

I would like to understand the following equality and inequalities:

a) $$\frac{1}{4} |\nabla_i H \ h_{kl} – \nabla_k H \ h_{il}|^2 = \frac{1}{2} (|A|^2 |\nabla H|^2 – |\nabla^i H h_{il}|^2)$$;

b) $$|H \ \nabla_i h_{kl} – \nabla_i H \ h_{kl}|^2 \geq \frac{1}{2} \sum\limits_{i=1}^{n-1} \lambda_i^2 |\nabla H|^2$$;

c) $$\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \frac{|\nabla H|^2}{2(n-1)|A|^2} \geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2 \frac{|\nabla H|^2}{n(n-1)|A|^2}$$.

My thoughts:

$$a$$ and $$b$$) I just consider use normal coordinates, but this doesn’t helps me because the right hand side in $$a$$ and $$b$$ would be zero.

$$c$$) I just try to prove that $$\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n \lambda_i^2 \lambda_j^2 \geq \left( \sum\limits_{i,j=1, \ i < j}^n \lambda_i \lambda_j \right)^2$$, but I can’t prove this because I don’t know if all eigenvalues are non-negative. Indeed, I even don’t know if the $$H > 0$$ because I didn’t see the hypothesis that the hypersurfaces is mean convex on the paper until this lemma.

## Signed curvature of catenary involving turning/tangential angle

Suppose we want to find the signed curvature of the catenary $$\gamma(t)=(t,\cosh t)$$where $$\mathcal{k}_n=\frac{d\phi}{ds}$$ and $$\phi(s)$$ is the turning angle of $$\gamma$$ such that$$\dot\gamma(s)=(\cos\phi(s), \sin\phi(s))$$

We proceed: $$s=\int_{s_0}^{s}|\dot\gamma(t)|dt=\int_{s_0}^s\sqrt{1+\sinh^2t}dt=\sinh t$$ so if $$\phi$$ is the angle between $$\dot\gamma$$ and the $$x$$-axis, then $$\tan\phi=\sinh t=s\implies\sec^2\phi\frac{d\phi}{ds}=1\implies k_s=\frac{1}{1+s^2}$$

Can someone explain how we proceed in that last line of calculation? Why does $$\tan\phi=\sinh t=s?$$

## Curvature of plane curves

My text defines the curvature of a plane curve as $$<\ddot{x},N>$$ where $$N$$ is the normal to the normalized tangent of $$x$$ and $$x$$ is the curve. I thought the $$\ddot{x}$$ also was perpendicular to $$x$$, making this projection kind of odd. Can someone see where I go wrong? Isnt projectuon of parallel lines a wierd thing?

## Example of a Manifold which has One Non-zero Component of Ric corresponding to Scalar Curvature

I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $$R$$, I can find a manifold such that the Ricci tensor has all zero components except for one component which takes the value $$R$$.

I feel like this can be achieved using a warped product of two metrics to separate out one coordinate and then just solve the differential equations so that the first coefficient vanishes, but obviously the coefficient of the non-zero component needs to be $$R$$.

## What curve minimizes distance from the origin, given length and total curvature?

Let $$\textit{F}$$ be the family of $$C^1$$ curves in $$\mathbb{R}^2$$ of fixed length $$\bar{l}$$ and fixed tangent’s turning angle $$\bar{k}$$.

What are the curves in $$\textit{F}$$ minimizing the Euclidean distance between the starting and the ending point? Is the arc of circle of proper radius one of those?