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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Does any compact boundary become a minimal hypersurface iff each of the boundary component is a minimal hypersurface?

There are researchers who made the following assumptions in their paper on the Penrose inequality: Let $M$ be a complete, connected Riemannian $3$-manifold and suppose that the boundary $\partial M$ ...
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Proof that every analytic function describes a minimal surface

Is my proof corect? Hypotesis: "Every analytic function describes a minimal surface." Proof: Every analytic function satisfies the Laplace equation: $\nabla^2 u = \frac{\partial^2 u}{\...
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Ants, cylinders and shape operator

For an oriented surface in $\mathbb{R}^3$ the curvature tensor can be expressed as $$ R(X,Y)Z = \langle L(Y),Z \rangle L(X) - \langle L(X),Z \rangle L(Y) $$ where $X,Y,Z$ are vectors fields on the ...
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Lorentzian manifolds of constant curvature

I know that Riemannian manifolds of constant curvature are locally isomorphic to the spherical, flat or hyperbolic space depending on the sign of the sectional curvature. Reading Hawking and Ellis' ...
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I don't understand why the curvature of a function with Hessian $H$ in a direction $p$ is $p^T H p$

I am reading a book that claims (without proof) that the curvature of a function $f(x)$ along a direction $p$ is obtained by projecting the product of the Hessian $H$ and $p$ onto the direction $p$ i....
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Level-set formulation of inverse mean curvature flow (IMCF)

I'm thinking about an equivalent formulation of IMCF proposed by Gerhard Huisken and Tom Ilmanen in their 2001 work THE INVERSE MEAN CURVATURE FLOW AND THE RIEMANNIAN PENROSE INEQUALITY: Let $M$ be a ...
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Totally antisymmetric Riemann tensor

Is it possible for a Riemannian manifold to have a non-vanishing Riemann tensor that is totally antisymmetric in the four indices? Of course, the antisymmetry would imply that the Ricci tensor ...
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Does Curved Edge exist for a smooth infinitely long right circular Cylinder?

This question is in the continuation of this question. As it is cleared from the comments of the respective question that an infinitely long cylinder which is also a right circular, is a smooth $3$D ...
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Positive Gaussian Curvature

So my quastion is who does one show that if $M\subseteq \mathbb{R}^2$ is a manifold with positive gaussian curvature (G.C) at every point and if a curve $\gamma\subseteq M$ is such that $\gamma'\neq0$ ...
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Constant Curvature Reparametrization

I am getting into differential geometry and came up with the problem which is unexpectedly difficult to solve. Suppose we have a curve $\alpha(t):[0,L]\rightarrow \mathbb{R}^2 $ parametrized by ...
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Is this a correct way to derive the Riemann curvature tensor?

I wanted to see if I can use a simple idea and derive the Riemann curvature tensor by only using the covariant derivative, because I somehow got the impression that the derivations I read in several ...
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Determinant of a Frenet curve

I have this problem: Let $c$ be a Frenet curve in $\mathbb{R}^n$. Show that $ \operatorname{Det}\left(c^{\prime}, c^{\prime \prime}, \ldots, c^{(n)}\right)=\prod_{i=1}^{n-1}\left(\kappa_i\right)^{n-i}...
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Integrated Curvature on Manifolds in SageMath

Is it possible to integrate the Ricci curvature scalar on an arbitrary $n$-dimensional manifold in SageMath? Just straight up: $$\int R dV$$ Assume that I know how to get the Scalar curvature a la ...
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Finding maximum curvature of $Qe^{-ax}$?

Background I have the following function: $H(x)=Qe^{-ax}$, where all parameters are positive. Using Mathematica, I used the following formula to calculate $H(x)$'s curvature: $H_{C}=\frac{|H''(x)|}{\...
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If $S = P^{-1}(0)$ is a surface, show $\int |K|\leq 4\pi C(d)$

Let $P\in \mathbb{R}[X,Y,Z]$ be an irreducible polynomial, such that $S=P^{-1}(0)$ is a non compact regular surface. If $K$ denotes the Gaussian curvature, prove that:$$\int_S |K|\leq 4\pi C(d)$$ ...
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Does a geodesic locally minimize the curvature?

Does a geodesic somehow minimize the local curvature? Lets say we have a smooth geodesic $\gamma$ with constant speed on a surface $S\subset\mathbb{R}^n$ and with $\gamma(t)=x$, then every other ...
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Rewriting Laplacian powers $\Delta^l$ on a boundary using directional derivative along the normal

How would one go about rewriting $\Delta^l u$ on a boundary $\partial\Omega$ in terms of $\partial_n^{2l}$, where $n$ is the normal of $\partial\Omega$? I am asking since I saw in remark 2.11 of the ...
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Verifying the Bakry-Émery criterion on spheres

Suppose that I am interested in a probability measure $\mu$ defined on the sphere $\mathbb{S}^{d-1} \subseteq \mathbb{R}^d$, whose density with respect to the normalised surface measure can be written ...
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Differences in mean curvatures for permuted coordinates of a manifold

Below is an example demonstrating the differences in mean curvatures and one other quantity, and below is my question. Please comment if you need me to explain in more detail or clarify notation, or ...
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Reconciling different expressions for Riemann curvature tensor

[Note: This has been crossposted to Physics SE, but I haven't found a thourough explanation there so far, so I'm posted the question here as well] I'm using Einstein's summation convention throughout. ...
2 votes
1 answer
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Which Riemann tensor should I choose?

Apparently there are two conventions for the (1,3) tensor called the Riemann curvature and they give opposite functions $$T_p M \times T_p M \times T_p M \to T_p M :$$ $$ (x,y,z)\mapsto R(x,y)z.$$ ...
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Convexity and positivity of the second fundamental form

Let $\Omega$ be a smooth bounded open set in a complete Riemannian manifold $(M^n,g)$, $n\geq 2$, and denote by $N$ the outward unit normal for $\partial \Omega$. Is it true that $\Omega$ is (strictly)...
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Curvature of the graph of a function defined on a sphere

I am not an expert in differential geometry and was wondering: Given a function $h:\mathbb S^{n-1} \longrightarrow \mathbb R $ smooth, the graph of $h$ is an $(n-1)$-dimensional submanifold of $\...
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Is there any natural way to turn flat space with torsion into curved space without it?

Say we have a smooth function $f: \mathbb{R}^n \rightarrow SO(n)$, whose value is asymptotically zero in all directions. In other words, we're specifying a rotation at every point of space, but ...
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Why is curvature scale $O(1)$?

The question comes from book "NETLAB: Algorithms for Pattern Recognition". In section 2.3.1 on page 42-43, the book first introduces bracketing interval: A bracketing interval is a triple $...
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Question about ratio of Riemannian densities

I am studying the paper "On the MCP property of metric measure spaces" by S. Ohta (https://ems.press/content/serial-article-files/1508) and I have a question regarding a passage at page 813 ...
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Can a spline be constructed given start and end points, normalized tangent vectors, and curvatures?

I want to create a smooth curve that starts at point $P_0 = (Px_0, Py_0)$ with normalized tangent $T_0 = (Tx_0, Ty_0)$ (where $\sqrt{Tx_0^2 + Ty_0^2} = 1$) and with curvature $κ_0$ (where positive ...
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Problem with a Curvature Calculation on $S^3\rightarrow S^2$

Sorry this going to be mildly long, I am having trouble obtaining what I think I should when I pull back the curvature form $S^3$ to $S^2$ via a local section, and would like to see if someone can ...
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Calculation of Riemann Curvature Tensor

I am reading the book: "Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers" (P.M.Gadea, J.Munoz Masque) Problem 6.5.1. Page 252-253: Find the Riemann ...
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Why does a path homotopy between distinct geodesics must pass through a long curve?

I'm working on problem from Do Carmo's "Riemannian Geometry" (Ex. 1, chapter 10). The problem, called Klingenberg's Lemma, puts a lower bound on the length of some curve in a homotopy, ...
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Can we think of the Riemann curvature tensor as a tensor-valued 2-form and integrate it over a surface?

The usual physical interpretation of the Reimann tensor $R^\mu_{\ \ \nu \rho \sigma}$ at a given point $p$ on a manifold $M$ is that it inputs an infinitesimal vector $v^\nu$ and two other ...
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Torsion on a flat connection (Geometric Intuition).

Start with the 2-sphere $\mathcal{S}^2$ with the standard $(\theta, \phi)$ chart ($\theta = 0$ is the North pole etc) and metric: $$ds^2 = d\theta^2 + \sin^2 \theta d\phi^2$$ Remove the two poles from ...
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How does curvature relate to angle measurement in hyperbolic geometry?

This question is about the relationship between curvature and angle measurement in hyperbolic geometry... Specifically, I am trying to understand the following excerpt from pp. 489-490 of Greenberg's ...
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Prove that vector tangent to a curve always maintains the same angle, so the ratio between curvature and torsion is constant

I have a 3D curve parametrized by the arc length, $r(s)$, which I compute the curvature $\kappa$ and de torsion $\tau$ as: \begin{equation} \kappa(s) = ||\frac{dT}{ds}|| = ||r''(s)||\\ \tau (s)...
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Calculating scalar curvature of a warped product $I \times N^n(k)$

I'm trying to work out the details of this example by R. Bryant. He considers a Riemannian manifold $(N ^n, h)$ of constant sectional curvature $k$ and constructs the following quadratic form on $\...
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Curvature of Connection [exercise proofcheck]

I tried to solve the following question, but I get the feeling I am doing a few things wrong, but cannot really find out what. So the exercise is: Let $\alpha, \beta \in \Omega^{1}(M)$ for a manifold $...
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Curvature defined as exterior covariant derivative of connection?

In Gauge Theory and Variational Principles by David Bleecker, the curvature of a $\mathfrak{g}$-valued 1-form connection is defined as $\Omega^\omega:=D^\omega\omega=(d\omega)^H$, where $\omega(d\...
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Derive formula for conformal curvature

So I'm working through Needham's Visual Differential Geometry and Forms, and what is suggested as a simple exercise is to reduce the general Gaussian curvature formula $$K = -\frac{1}{AB}\left(\...
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Integration of Ricci curvature on a compact manifold.

$\mathbf {The \ Problem \ is}:$ Let, $(M,g)$ be a compact Riemannian manifold and $X\in \chi(M).$ Show that $\int Ric_g(X,X)\mu_g=\int ((tr(\nabla_.X))^2-tr(\nabla_.X\circ \nabla_.X))\mu_g.$ $\mathbf {...
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Realization of the $2$-form as the curvature of some bundle.

Suppose $M$ be a manifold, and $\omega$ be any smooth $2$-form. My question is about the existence of the Lie group $G$ which satisfies that there is always a principal $G$-bundles $P$(or its ...
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Chern class of a hypersurface in $\mathbb{C}P^3$

Let $X=\{[z_0,z_1,z_2,z_3]\ \big{|}\ [z_0,z_1,z_2,z_3]\in\mathbb{C}P^3,z_0^4+z_1^4+z_2^4+z_3^4=0\}$. $c_1(X)$ is the first Chern class of $X$. Prove that $c_1(X)=0$. $\textbf{My try}$: It's easy to ...
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No Nash-Kuiper theorem for $\mathcal C^2$ isometries

Reading about the Nash-Kuiper theorem, I found the following statement in Y. Eliashberg and N. Mishachev's book Introduction to the $h$-Principle: Is there a regular homotopy $f_t:S^2\rightarrow\...
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Minimizing the maximum curvature: a real-life, liquid-filled pipe question [closed]

I've searched Stack Exchange for this question, but have only found Bezier curves which I don't think is what I'm looking for. I have a real-world problem where I need to design a system of water-...
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Radius of Curvature for Cam Profile

In the design of a radial design cam-follower mechanism, the radius of curvature of the cam profile is often used to identify areas of high contact stress. The pitch circle of a cam is defined by ...
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Ambiguity in trace of curvature 2-form matrix

My question is about computing the first Chern class from the trace of curvature 2-form matrix via the Chern–Weil theory. First Chern class is given by $$ c_{1}=\left[{\frac {i}{2\pi }}\operatorname {...
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Why non zero wedge term in curvature form?

The definition of the curvature form ( from Wikipedia ) is: $$ \Omega = d \omega + \frac{1}{2} [ \omega \wedge \omega ] $$ I am puzzled about the rightmost term. According to what I have learned so ...
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Gauss Equation in Coordinates

Consider a $(d+1)$-dimensional pseudo-Riemannian manifold $(\mathcal{M},g)$ and a Riemannian hypersurface $(\Sigma,h)$, i.e. $\Sigma$ is an embedded submanifold of $\mathcal{M}$ of codimension $1$ ...
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Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces

Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
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Expression of a metric tensor with constant curvature on surface

Consider the plane $\mathbb{R}^2$ together with a metric $\mathsf{g}$. Choosing suitable local coordinates $(x_1,x_2)$, $\mathsf{g}$ takes the form : $$ \mathsf{g}=r(x_1,x_2)\left((dx_1)^2+\epsilon (...
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Compact surface with Gaussian curvature 0 everywhere

I'm trying to give an example of a compact surface $S\subset\mathbb{R}^3$ with constant zero Gaussian curvature for all $p\in S$. I know that a plane would have constant zero Gaussian curvature, but ...

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