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 derivative of a simple closed curve $\alpha$ with certain properties injective?

Let $\alpha : I \to \mathbb R^2$, where $ I \subset \mathbb R$, be a simple closed curve with positive speed and curvature, does it follow that $\alpha’$ is also injective, as a vector field on $\...
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Does every smooth unit-speed curve in $\Bbb R^3$ have a well defined torsion? Explain.

The torsion at $\gamma(s)$ for a unit-speed curve $\gamma$ in $\Bbb R^3$ is the value $\tau \in\Bbb R$ such that $\textbf{b'(s)}= -\tau(s)\textbf{n(s)}$ where $\textbf{b(s)}=\textbf{t(s)}\times\...
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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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Visualize manifold from the metric and extrinsic curvature?

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 +\...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)}\...
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Different definitions of mean curvature

Wikipedia gives the following definitions of the mean curvature $H$ at a point $p$ on a surface $S$: $$ (1) \;\;\;\;\;\; H = \frac{1}{2\pi} \int_0^{2\pi} \kappa(\theta) d\theta$$ where $\kappa(\theta)$...
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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 ...
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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\}...
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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 ...
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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{\...
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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{\...
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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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Different definitions of total curvature?

I'm looking at this paper 'Boundary conditions at a liquid-vapor interface' by Prosperetti 1979 and on the second page this definition is given for the total curvature: The total curvature $\mathscr{C}...
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General way to calculate Gaussian Curvature in 4D

I am kind of confused by the vast number of formulas for computing the Gaussian Curvature. Having a metric tensor / an expression for the line element in 4D (e.g. $t,x,y,z$ or in spherical coordinates ...
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Is there a finite boundariless pseudosphere without edges and vertices?

A sphere represents finite boundariless surface with constant positive curvature, while a Clifford's torus represents finite boundariless surface with constant zero curvature. Both of these surfaces ...
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Geometric interpretation of Ricci tensor acting on two vectors?

I've been looking into the geometric interpretation of the Ricci tensor, and the standard idea is that, for example in 3D, $Ricci(u,u)$ corresponds to the rate of change of volumes in the direction u ...
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Proving Geometric definition of divergence of vector field as given by Tristan Needham

In page-479 of Visual Complex Analysis, Tirstan Needham derives the flux of a vector field in Geometric form: Here $z$ is the point to which the shaded region $R$ will ultimately be collapsed in ...
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Is the conformal Killing factor always an eigenvalue of the Laplacian?

Suppose $(M, g)$ is a pseudo-Riemannian manifold and $\xi_{\mu}$ is a conformal Killing field, i.e. $$ \nabla_{\nu} \xi_{\mu} + \nabla_{\mu} \xi_{\nu} = \kappa g_{\mu \nu} $$ for some smooth ...
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Elliptic point of surfaces

Why does a point $p$ in a surface $S \subset \mathbb{R}^3$ such that the Gaussian curvature is positive is called elliptic point?
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Riemann Curvature of Schwarzschild with curvature 2-forms

I have been studying Chapter 14 of Misner, Thorne, Wheeler's Gravitation, in particular the method of computing curvature using exterior differential forms. Is there a reference that computes the ...
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Proving the curvature formula for an arbitrary planar curve using perpendicular bisectors

Given an arbitrary (i.e. not necessarily arc-length parameterised) planar parametric curve $C(t) = \Big(x(t), y(t)\Big)$, I'm looking to prove the formula for its (signed) curvature $$\kappa = \frac{x'...
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Can coefficients of a weighted average be chosen to guarantee maximal curvature?

Problem setup: We have a set of samples $\{(X_1, Y_1), (X_2, Y_2), \dots \}, i = 1, \dots, n$ Based on these samples, we are trying to find an estimator $\hat{y}(x)$ which minimizes some error term. ...
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Curvature for Markov chains in discrete spaces

I'm currently reading some papers regarding curvature for markov chains in disrecte spaces, which is not my main field in mathematics and got a question. In the paper of Frank Bauer, Paul Horn, Yong ...
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Are there alternate geometric interpretations of the Riemann tensor based on its symmetries?

In Riemannian geometry, when we think of the Ricci tensor merely as $Ric(\vec a, \vec b)$, then it's hard to describe it other than saying it's the contraction of the Riemann tensor. But when we ...
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Changing the signs in the metric and the curvature

Suppose we have two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ and consider their product $C:=B\times F$ endowed with the metric tensor $g_C:=-\pi_B^*(g_B)+\pi_F^*(g_F)$, where $\pi_B$ and $\pi_F$ ...
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How to find regularity, the signed curvature, and arc length of $\gamma(s)+\epsilon \pmb{n}(s)$

Question: Let $\gamma:[a,b]\to \mathbb{R}^2$ be a regular plane curve. For a constant $\epsilon\in \mathbb{R}$, define $\tilde{\gamma}:s\mapsto \gamma(s)+\epsilon \pmb{n}(s)$. Show that i) If $|\...
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