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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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Can I assume a horizontal vector is a horizontal lift?

I am reading a proof that the curvature 2-form is given by $(F_A)_p(v,w) = dA_p(v,w) + [A_p(v),A_p(w)]$ where $A\in \Omega^1(P;\mathfrak{g})$ is a connection 1-form on $P\xrightarrow{\pi}M$. One of ...
Wyatt Kuehster's user avatar
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Tangent developable is locally isometric to an open set in ℝ²

I'm trying to solve this problem about tangent developable from a differential geometry exam $\sf 1996\ Q3$: My working: First part $\ldots\ldots$deduce that $Σ$ is ruled (that is, each point of $Σ$ ...
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Understanding Chern Classes [closed]

Let $E$ be a vector bundle over a complex manifold $X$ with curvature form $F_{\nabla}$ of the Chern connection. The Chern classes are defined by: $$ \det\left(\frac{i}{2\pi}tF_{\nabla} + I_n\right) = ...
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Natural parametrization to paramatric [closed]

I need to find radius-vector of the curve having the equation of its curvature (specified in natural parameter s): $$ k(s) =\dfrac{1 }{s^2 \sqrt{1 - s^{-2}}} $$ Do you know any soft capable to solve ...
Роман's user avatar
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Relationship between hessian matrix and curvature [closed]

I am taking vector calculus this semester, and while researching about Hessian matrices for a project, I encountered this formula. enter image description here Could anyone explain how it is derived, ...
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Umbilical points under isometries

In a certain question I have been asked to say if it is true that given a surface $S$, if a point $p$ is umbilical then under an isometry $f$ with another surface $S'$ the transformed point $f(p) \in ...
Emmy N.'s user avatar
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Holonomy along a circle of constant latitude and Gaussian curvature

I ran into a problem that I cannot reconcile by myself when studying the relation between holonomy and Gaussian curvature. In particular, let $S$ be a sphere of radius $1$ parametrized as usual: $$\...
Sean Ian's user avatar
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Checking an error in a Differential Geometry problem on curvature and diffeomorphism types of compact surfaces

I am currently working through the following old exam problem. I have reached part (d)(i) but I believe the result required still is not guaranteed even with the new hypotheses. For example, if we ...
John Robertson's user avatar
2 votes
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Gaussian curvature of Hyperbolic Plane.

I'm studying about Gaussian curvature from Elementary Differential Geometry - Barret O'Neill, Revised Second Edition. More precisely, in geometric surfaces. In section 7.2, example 2.5, it comes up ...
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Radius relation of four consecutively touching circles $O_7,O_2,O_3,O_6$ with common tangent circles $O_1,O_4$

A construction of four consecutively touching circles $O_7,O_2,O_3,O_6$ that all touches circles $O_1,O_4$: Triangle $O_1O_2O_3$ has outer Soddy circle $O_4$, so that the circles $O_1,O_2,O_3,O_4$ are ...
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Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
zeta space's user avatar
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Discrepancy in Calculating Scalar and Sectional Curvature in 2D Geometry

I've been working on a problem related to curvature in two-dimensional geometries and found a discrepancy between my calculations of scalar and sectional curvature, which seems to contradict the known ...
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a cool optimization problem involving cubes surfaces and volumes

Consider a codimension one surface of revolution $S$ and an embedding $e:S \hookrightarrow X^n$ for $X^n=[0,1]^n$ with points $p,q$ elements of $\partial X^n$ where $\partial X^n=X^n-(0,1)^n$ for $\...
zeta space's user avatar
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How do I calculate an orthonormal basis, with zero-valued second fundamental form, of a smooth parametric surface at any given point?

Main Question How do I find an orthonormal basis $X_1$, $X_2$ of tangent vectors at a point $P$ on a surface $S$ such that $I\!I(X_{1}, X_{2}) = I\!I(X_{2}, X_{1}) = 0$, where $I\!I$ is the second ...
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Formula in Chern's Paper Curvatura Integra

I'm currently reading ON THE CURVATURA INTEGRA IN A RIEMANNIAN MANIFOLD from Chern. And I have trouble to understand the calculation for formula (6). I took the formula for $d\phi_k$ and then I ...
SteuerWB's user avatar
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A graph with radius of curvature $≥1$ can't have more than 2 distinct real intersection points with a circle of radius 1

Is the following true? If the graph of a continuously twice differentiable function $y(x)$ and the radius of curvature $|\frac1\kappa |$ is $\gt 1$ at all points on the graph (e.g. $y=\sin(x),x\in(0,\...
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Examples of noncomplete, simply connected Riemannian manifolds with nonpositive sectional curvature

I will be giving a talk on the Cartan-Hadamard theorem next week (in the context of a learning/reading group), which states (in the form I will be presenting) that a complete simply connected ...
Maxman013's user avatar
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Verifying that $M$ has curvature $-1$ iff $f''=f$ when $g=dr^2+f(r)^2d\theta^2,M=[0,\infty)_r\times S_\theta^1$

I am currently at a difficult position, because I have to check some definitions/examples regarding hyperbolic surfaces, but I have not taken a proper course on Riemannian manifolds or surfaces in the ...
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Determining embedded Parameterization in 3D of a 2D manifold when only the metric is given, by exploiting symmetries

I have the second fundamental form $$E(h,\gamma) = \frac{1}{16} \frac{h \gamma^2}{ (h^2-1) (h^2+\gamma^2-1)^{\frac{3}{2}}}, $$ $$F(h,\gamma)= -\frac{1}{16}\frac{ \gamma}{ (h^2+\gamma^2-1)^{\frac{3}{2}...
prikarsartam's user avatar
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Do we need to use wedge product or tensor product for Curvature form in local frame?

From this note, On section $4.1.1$ we have, If $\{e_\alpha\}$ be a local frame of a vector bundle $\pi:E\rightarrow X$ then there exist a section $\Theta\in C^\infty(M,\Omega^2_N\otimes E)$, called ...
WhyMeasureTheory's user avatar
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Self duality of a connection is invariant under a gauge transformation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $P\to M$ be a (smooth) principal $G$-bundle over an oriented Riemannian smooth 4-manifold $(M,g)$. Let $E=P\times_{\text{Ad}}\mathfrak{g}...
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Curvature of a product of two functions

Can we estimate how the curvature of a function changes if "bent" over another function? Example: $y_1(s)$ - is sinusoidal function, e.g., $a\cdot \sin(b\cdot s), \; s \in [0, L]$ $y_2(t)$ -...
Roman Zh.'s user avatar
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how to find circle of curvature centre point

For this question I am asked to find the radius and centre-point of the circle of curvature for the following function: $−7.87e^{2.65x}$ I calculate the radius correctly with the formula: $R =\frac{...
reira osaki's user avatar
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How to find the circle of curvature centre

For the following question I am asked to find the radius for circle of curvature for the function: $-0.18e^{4.88x}$ I found the radius 1.681459915, by using the formula: R =1/ρ and this was correct ...
reira osaki's user avatar
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Probability measure satisfying curvature-dimension

Assume that the non-compact $n$-dimensional Riemannian manifold $M$ satisfies a curvature-dimension $\mathrm{CD}(K,N)$, $K \le 0$, with respect to a measure $\mu = e^{-V} d\mathrm{Vol}$; i.e. \begin{...
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Computations of tensor on a Riemannian Manifold.

I am trying to get some practice in computing tensors in Riemannian and Pseudo-Riemannian manifolds. Is there somewhere that I can check my results? For example right now I am computing the ...
S_d_pap's user avatar
6 votes
1 answer
171 views

Help Antie evaluate Gauss curvature of a smooth surface using ruler and a protractor

Antie, a smart ant living on a smooth surface $S$ of $\mathbb{R}^3$, would like to evaluate the Gauss curvature $K$ at a certain point $P\in S$. Antie is aware of Gauss Theorema Egregium, according to ...
Nikolaos Skout's user avatar
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Curvature of "music-scale"

Lately I am thinking about music from a mathematical point of view. Let us consider $\mathbb{H} =\mathcal{L}^2([0,T])$ as the Hilbert space of functions that represents the time-intensity plot of a ...
Andrea Marino's user avatar
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Clarification on Sign in Mean Curvatures of Parallel Surfaces

For my Differential Geometry class, I've encountered an issue with a problem from do Carmo. Although I'm aware that similar questions have been previously addressed, my issue differs. I understand how ...
PaulNord90's user avatar
1 vote
1 answer
50 views

Finding the second derivative of $1.51x^2 + y^2 = 1 + 0.71x^2y^2$ to calculate the curvature

Consider the equation $$1.51x^2 +y^2 = 1 + 0.71x^2y^2.$$ In this question you will calculate the curvature, $\rho$. Evaluate the derivative at the point described — you should get decimal numbers $$x= ...
Lollipop 's user avatar
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The integral of the absolute value of the Gaussian curvature of a compact surface

I want to prove the following theorem: Let $S$ be a compact surface, and $N:S\rightarrow \Bbb{S}^2$ the Gauss map, then we have $$ \int_{S} |K| \,dA = \int_{\Bbb{S}^2} \#N^{-1} \,dA $$ where $K$ is ...
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1-7 Exercise 4 Do Carmo Differential Geometry

The question is: Let $C$ be a plane curve and let $T$ be the tangent line at a point $p \in C$. Draw a line $L$ parallel to the normal line at $p$ and at distance $d$ of $p$ (Fig. 1-36). Let $h$ be ...
Shaw Chan's user avatar
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A dilation's effect on signed curvature

Suppose $k$ is the signed curvature of a plane curve $C$ (parametrized by $\gamma$, say) expressed in terms of its arc length. Show that, if $C_a$ is the image of $C$ under the dilation mapping $\vec{...
Bifton Mifts's user avatar
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Possibility of superposition of nonpositive curvature curves produces positive turning

Suppose a finite set of plane curves parameterized by $t$: $p_i(t)=(x_i(t),y_i(t)), i=1...N$, satisfy boundedness: $|p_i(t)|\leq 1\ \ \forall i$ smoothness: $p_i(t)$ differentiable to any order, $\...
George C's user avatar
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Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
Cris's user avatar
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127 views

Relation between curvature form and the Riemann tensor

Let $ (M, g) $ be a Riemannian manifold and $ \omega $ a connection on the tangent bundle of $ M $ at a given point. Then the curvature form is the $\mathfrak{g} $-valued 2-form $ \Omega \in \Omega^2(...
Tomás's user avatar
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Sign discrepancy between two different definitions of ricci curvature

In my differential geometry course, the $(1,3)$-riemann curvature tensor $R$ is defined by $$R(X,Y)Z:=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ and the $(0,4)$-riemann curvature tensor Rm ...
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1 vote
1 answer
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Curvature of a special curve

In the following problem I am asked to find the curvature of a curve. The problem is the following: \ Given the curve $\alpha(t), \alpha: \mathbb{R} \rightarrow \mathbb{R}^3$, which is parametrised ...
Emmy N.'s user avatar
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1 vote
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Shape operator and Gaussian curvature of a hypersurface: using surface and embedding connections

(I've been self-learning differential geometry for some time now, and math.stackexchange has been of sooo much help, that I thought I might get my answer here.) From Lee's "Introduction to ...
Ale's user avatar
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1 answer
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ring meniscus at cylinder

I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry ...
creillyucla's user avatar
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1 answer
40 views

How does the riemann tensor symplify to normal components in the second variation formula?

I have a doubt in the last step of the derivation of the second variation formula for the length functional (Lee's intro to riemannian geometry) (theorem 10.22 see the proof below) Why is it true that ...
some_math_guy's user avatar
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How do I find constants so that the curvature of an oscillating funtion is between them?

I am currently struggling with how to answer this question: "Consider the planar curve $y = a\sin(bx)$ where $a, b$ are non-zero real numbers. Find constants $A$ and $B$ so that $A \le k \le B$ ...
Jarvis's user avatar
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Extrinsic curvature of a curve

I have a simple question. Curve in ${R}^{1,1}$ The 2 dimensional Minkowski space. \begin{equation} \gamma:[0,1] \longrightarrow R^{1,1}, \end{equation} How can I calculate the extrinsic curvature of ...
LuVa's user avatar
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9 votes
4 answers
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Motivation for the definition of curvature of a plane curve

I am seeking a motivation for the definition of the curvature of a plane curve. How did people come with the idea of the definition of the curvature? Below I am more specific. The fundamental theorem ...
ghreis's user avatar
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Is the metric in normal coordinates constant for flat manifolds?

Simple question. Let $(M,g)$ be a flat manifold (i.e. a Riemannian manifold with vanishing Riemannian curvature). Is it true that the metric $g_{ij}$ in normal coordinates around each point is ...
Filippo's user avatar
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Calculations involving the Ricci tensor on $S^2$.

I am trying to do some calculations with involving the Ricci tensor, but I am off by a factor 2. Here's my problem: The manifold is $S^2$ with the standard metric $g=e^{2\phi}\delta_{ij}$, with $\phi=\...
user3646557's user avatar
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0 answers
45 views

Normal of curvature of curves in a surface

Let be $ S=\{ (x,y,z) \in \mathbb{R}^3| z=y^2-3x^2 \} $ Determine the normal curvature at time $t=0$ of curves parametrized by arc length $ \gamma: (-1,1)->S$ with $\gamma(0)=(0,0,0)$ Find two ...
Andreadel1988's user avatar
1 vote
0 answers
104 views

Isometry of surface

Let $S$ be a set of points $ (x,y,z)\in\mathbb{R^3}$ that satisfy the equation $x^3+y^2+z^2=1$, $p_1=(0,1,0)$ and $p_2=(0,0,1)$ points in $S$ with the same gaussian curvature,prove that $f: S \to S$ ...
Andreadel1988's user avatar
3 votes
2 answers
168 views

Principal curvature of a surface defined by an equation

Let be $S$ a set of points $ (x,y,z)\in\mathbb{R^3}$ that satisfy the equation $x^3+y^2+z^2=1$. Calculate the main curvatures and Gaussian curvature at the points $p_1=(1,0,0)$, $p_2=(0,1,0)$ e $p_3=(...
Andreadel1988's user avatar
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Normal curvatures of differentiable curves on a surface with zero Gaussian curvature

Let $S$ be a surface, and $p$ a point on $S$ with zero Gaussian curvature. Can there exist two curves $\gamma_1, \gamma_2 : (a,b) \to S$ with $\gamma_i(0)=p$ and parametrized by arc length, such that ...
Andreadel1988's user avatar

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