# 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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### How to interpret "Ricci-tensor represents volume gain"?

The Ricci-tensor is told to be the volume gain in comparison to Eucledean (Lorentzian) space (wikipedia-page on Ricci-Tensor). Question part 1: Is that the volume change of the manifold or of an ...
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### Gauß-Bonnet for non-compact manifolds

Is there a reasonable approach to extend the Gauß-Bonnet theorem to non-compact manifolds? and use this to distinguish between i) the 2-sphere and ii) the punctured 2-sphere?
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### Having a closed and periodic curve $C$, in the unit sphere $\mathbb{S^2}$, calculate $\int_{C}\mathcal{K}_g(s)ds$

Let us have a closed and periodic curve $C$, in the unit sphere $\mathbb{S^2}$.(T understand better, visually the curve is like this: ~ , around the equator of the sphere). I have to calculate the ...
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### Is the Principle Normal vector orthogonal to the tangent vector of a curve?

Given that the principle normal vector for a unit-speed curve $\lambda$ in $R^3$ is $\textbf{n}= \frac{1}{k}\textbf{t'(s)}$ where k is our curvature. Could anyone explain the intuition behind the ...
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### Curvature and position of tangent space

Let $M$ be a hypersurface of $\mathbb{R}^d$, $x_0 \in M$ and $T$ the tangent space of $M$ at $x_0$. Which of the following are true ? If the sectional curvature of $M$ at $x_0$ is strictly positive ...
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### Finding points on a parametric curve where curvature changes

I am a engineer working on Wankel motors, where a simil-Reuleaux triangle rotates eccentrically in a 8-shaped form: (from Wikipedia) Fascinated by this mechanism, I was studying the meaningful ...
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### Are tensors constructed such that one forms "act" on some complex vector field?

I am a physics student, trying to learn differential geometry. I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ ...
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### How to compute principal components for a curvature found given XYZ points?

I have a certain XYZ set of points that make up an object. I chose a random point and make the nearest radius analysis and find the neighbors. From these neighbors, I get the green pointcloud curve ...
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### Question from curvature. [closed]

Find the radius of curvature at any point (r,theta) for the following curve whose polar equation is given. Given Equation I have tried to solve the question but I got stuck in the middle of the ...
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### How does the following definition of hyperbolic arc length apply to distance problems? (Differential Geometry)

I have some doubts about the application of the following. Let be the regular surface $\mathbb{H} =$ {$(x,y,z)$ $\in \mathbb{R^3} | z=0, y >0$}. For each point $p =(x,y,0) \in \mathbb{H}$, define ...
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### Interpretation of the curvature of an helix

Consider the helix $h(t)=(a\cdot cos(t),a\cdot sin(t),bt)$ and the circle $c(t)=(a\cdot cos(t),a\cdot sin(t),0)$. I know that the curvature of them are: $k(t)_h=\frac{a}{a^2+b^2}$ $k(t)_c=\frac{1}{a}$...
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### Do the curvature properties of exotic spheres result in necessary new techniques for calculating arc length?

It was made clear to me in the post What answer does one get from integrating a Riemannian metric on a sphere for a great circle – does the metric’s non-flatness affect the answer? that a great circle ...
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### Solutions of Yang-Mills equation in the case of $G=U_1$

If $P\to M$ is a principal $U_1$-bundle, and $A$ is a connection on $P$, then it's curvature $F_A$ is a $2$-form with coeficient in $P\times_G\mathfrak{u}_1$, where $\mathfrak{u}_1$ is the Lie ...
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### Prove that if all geodesics of a connected surface $S$ has curvature 1 $S$ is contained in a sphere

I know that every geodesic curve of a connected surface $S$ has curvature 1. I want to prove that $S$ is contained in a sphere. I think that it can be useful to see that every point is umbilical to ...
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### Multi part question part 1: What is a metric? [closed]

I just watched this excellent YT video in which someone explained very clearly exactly what a metric is. This has led me to a few questions which I hope will help me to understand the math of GR. So ...
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### What is wrong with this surface? (principle directions)

The surface is specified, for simplicity, using an explicit function $f(x,y)$: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ The shape of this surface at point $[x=0,y=0]$ can be described by figure. ...
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### How does persistent homology detects curvature?

I am currently studying the paper Persistent homology detects curvature by Peter Bubenic (https://doi.org/10.1088/1361-6420/ab4ac0). I am stuck at a very fundamental idea of this paper. It claims that ...
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### What do the entries of the Riemann curvature tensor exactly represent? E.g. $R_{231}$

I understand that the Riemann curvature tensor is a rank-4 tensor, in other words a 4x4x4 cube of 64 numbers (which can be reduced because of symmetries etc.) The tensor represents the tidal force ...
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### Alternative expression for Riemann curvature tensor

There is the usual expression for the Riemann tensor $$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of ...
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