# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f$ ...
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### 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 $||\mathbf r'(t)|| = \sqrt{9t^{-4}+1}$. ...
### 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: ...