# 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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Is there some software or technique that would allow me to visualize a (2 dimensional) Riemannian manifold given its metric tensor and second fundamental form? For instance, given $$g =d\theta^2 +\... 1 vote 0 answers 28 views ### Point whose neighbourhood is contained in a certain sphere of radius r has Gaussian curvature greater than \frac{1}{r^2}? I am doing a question which asks one to show that is a compact, regular surface \Sigma has Gaussian curvature K < \frac{1}{r^2}, then \text{diam}(\Sigma) > r. The given solution is the ... 0 votes 0 answers 18 views ### Sectional Curvature of Convex Linear Combination of Metrics Suppose g_1 and g_2 are Riemannian metrics on a (say closed) manifold M. Denote$$|K_g| = \sup_{p \in M}\sup_{\Pi \subset T_pM}|K_p(\Pi)|,$$where K_p(\Pi) is the sectional curvature of the 2-... 0 votes 0 answers 27 views ### Growth of balls given an isometric, proper and cocompact G-action on X I am reading the paper "Elliptic and transversally elliptic index theory from the viewpoint of KK-theory" by G. Kasparov, and I have been trying to understand the following argument, which ... 0 votes 0 answers 34 views ### Calculating the sectional curvature of \mathbb{R}\mathbb{P}^n One way of showing this is using the canonical projection from \mathbb{S}^n to \mathbb{R}\mathbb{P}^n, and by showing that this is a local isometry, the sectional curvature of \mathbb{S}^n is ... 2 votes 0 answers 30 views ### Does a bounded harmonic function on a manifold with negative curvature have bounded Hessian I would like to find an example of a manifold (M,g) along with a non-constant harmonic function having bounded gradient and bounded Hessian. What came to my mind was that it is known (by the work of ... 0 votes 0 answers 19 views ### Distance between two closed convex bounded non-empty subsets in a CAT(0) space Let (X,d) be a complete CAT(0) space, where d is the metric on the space X. Being CAT(0) here means that for any x,y\in X, there is some m\in X such that for any z\in X, we have the ... 1 vote 0 answers 26 views ### Proving the inequality in the Fáry-Milnor Theorem is strict The Fáry-Milnor Theorem, as stated in Kristopher Tapp's Differential Geometry of Curves and Surfaces, states that, for a unit-speed simple closed (Tapp uses the convention that only regular curves are ... 2 votes 2 answers 41 views ### How can I find the curves whose curvature is k(s)=\frac{1}{as+b}? [closed] I'm trying to find the curves whose curvature is k(s)=\frac{1}{as+b}. I guess that they are Logarithmic Spirals. Parametric form of Logarithmic spirals: x(t)=ae^{(bt)}\cos(t) and y(t)=ae^{(bt)}\... 0 votes 0 answers 27 views ### Different definitions of mean curvature Wikipedia gives the following definitions of the mean curvature H at a point p on a surface S:$$ (1) \;\;\;\;\;\; H = \frac{1}{2\pi} \int_0^{2\pi} \kappa(\theta) d\thetawhere \kappa(\theta)... 0 votes 2 answers 31 views ### Applying to Gauss-Bonnet in the Angle-Sum Theorem and Circumference Theorem. I am studying about the Gauss–Bonnet Theorem in context of Riemannian Manifolds. I'm following the book "Introduction to Riemannian Manifolds" of John Lee. In the text, is defined the signed ... 2 votes 1 answer 84 views ### When does the Hessian on a Riemannian manifold vanish? The Hessian tensor of a smooth function f:M\rightarrow \mathbb{R} on a Riemannian manifold M with respect to the Levi-Civita connection \nabla is given, globally and in local coordinates \{x^i\}... 0 votes 3 answers 111 views ### Why is the Riemann curvature tensor a rank 4 tensor? The Riemann curvature tensor is defined as:  R(X,Y) = [\nabla_X, \nabla_Y]  when there is no curvature (no loss of generality in the question). If we expand this to coordinate notation, we get the ... 0 votes 0 answers 44 views ### Is a surface locally flat, when the curvature along a line is zero According to Spivak Vol. 2, p. 119 the following holds for a 2-dimensional manifold M = \left\{\left(x, y, f(x, y)\right): x, y \in \mathbb{R}\right\}: \begin{align*} K(x, y, f(x, y)) = \frac{\... 1 vote 1 answer 55 views ### Do metrics which agree on curvature everywhere give same geodesics? In page-38 of Visual Differential Geometry by Tristan Needham, the following equation for the metric-curvature formula is introduced: \kappa= - \frac{1}{AB} \left[ \partial_v \left[ \frac{\...
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 ...