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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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Definition of Tangentspace

Given a smooth manifold $M \ni p$ of dimension $m$ and smooth curves $\gamma,\delta:(-\epsilon,\epsilon) \to M$ with $\gamma(0) = \delta(0) = p$, I got a relation which identifies $\gamma$ and $\delta$ iff $D_0(\kappa \gamma) = D_0(\kappa \delta)$…
user42761
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Finding a space curve given some conditions on curvature and torsion

I have a space curve where the curvature is $\kappa$ and torsion $\tau = \kappa'$. An example of this would be a curve with curvature $\kappa = 1 - \cos s$ , $\tau = \sin s$. Is it possible to find the possible parametric equation of the curve?…
Vishesh
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Divergence of a smooth vector field

I was studying about the divergence of a smooth vector field in the book "Calculus of variations and harmonic maps"by Urakawa. For a smooth vector field $X$, the divergence is defined by $div(X)(p) := g(e_i,\nabla_{e_i}X)(p) ; p \in M$ wher…
user103005
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Skew line dense?

Please let me refer you to: Example 4.18. The skew line $f: \mathbb R \to S^1 \times S^1$ $$ f(t) = (e^{it}, e^{i\alpha t}). $$ If $\alpha$ is irrational then the image of $f$ is dense in $S^1 \times S^1$, so if $V$ is an open neighborhood…
user17182
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Differential Geometry Angle/First Quadratic Problem

Find the angle between the curves $v = 2u + 1, v = -2u +1$ on a surface with the first quadratic form: $E = 2, F = 1, G = 4$. I know I should probably use the $cos(\theta)=\frac{T_1(0)\cdot T_2(0)}{|T_1|\cdot|T_2|}$ however I'm unsure how to work…
user116900
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Given Constant Ratio of Torsion to Curvature, Show Tangent times Constant Vector is Constant

Let $r(t)$ be a unit speed curve such that for all $t$, $\frac{\tau(t)}{\kappa(t)}=\cot(\theta)$ for some $0 < \theta < \pi$. Show that there is a constant vector $a$ satisfying $T(t) * a = \cos(\theta)$ for all $t$ (Where $\tau$ is torsion,…
James Snyder
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Given Unit Speed Curve on a Sphere, Show the Curve has Constant Curvature

Let $r(t)$ be a unit speed curve on a sphere $x^2+y^2+z^2=R^2$. Show that the curve $c(t)=\int^t_0 r(u) du$ has a constant curvature $\frac{1}{R^2}$ I am still a little shaky with this stuff, so I don't know if I'm going about it the right…
James Snyder
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I need help with the derivation of the equation of a tangent line at a point on a curve, and the arc length parameter $s$.

I'm having difficulty seeing how under the arc-length parameterization the equation of the tangent line $$\frac{X-x}{dx}=\frac{Y-y}{dy}=\frac{ Z-z}{dz}=u$$ can be written as $$\frac{X-x}{x'}=\frac{Y-y}{y'}=\frac{ Z-z}{z'},$$ where $x'=dx/ds,\,…
Noor Aslam
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Connection form uniquely determined by linearly independent $\theta_1,\theta_2$?

I'm working through a tutorial for a differential geometry class. The question is: Consider the structure equations for a map $\bar x:\mathbb R^2\to\mathbb E^2$. Suppose that $\theta_1,\theta_2$ are everywhere linearly independent. Show that given…
PepperSausage
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Geodesic of Elliptic Hyperboloidv

I have the set $$M=\{(x,y,z)\in\mathbb{R}^3:z^2-(a^2x^2+b^2y^2)=R^2,\ z>0\}$$ and I have to write the differential equation that describe the geodesic curve and draw it. I used the parametrization $$ M=\left\{ \begin{pmatrix} x\\[1mm] …
MateAndres
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elements of the fundamental group and closed geodesics?

Let $(M;g)$ be a closed manifold . fix a point $x$ in $M$ and denote by $G=\pi_1(M;x)$ now let $\alpha$ in G i have 2 questions : 1- can $\alpha$ be represented by a closed geodesic ? 2- can $\alpha $ be represented by a closed minimizing…
user16712
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Proving a curve of a logarithimic spiral is 1/s cot(theta)

A unit-speed plane curve $\gamma$ has the property that its tangent vector $t(s)$ makes a fixed angle $\theta$ with $\gamma(s)$ for all $s$. Show that: (i) If $\theta = 0$, then $\gamma$ is part of a straight line. (ii) If $\theta = \pi/2$, then…
flunkinggrad
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rotation vector

If $t(t),n(t),b(t)$ are rotating, right-handed frame of orthogonal unit vectors. Show that there exists a vector $w(t)$ (the rotation vector) such that $\dot{t} = w \times t$, $\dot{n} = w \times n$, and $\dot{b} = w \times b$ So I'm thinking this…
ninja
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First variation of Area applied to minimal surfaces

I'm currently writing a project on minimal surfaces and have come across the First Variation of Area formula, however I'm finding it difficult to understand it's significance when understanding Minimal Surfaces. Would someone be able to…
Sarah Jayne
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Boundary of a $2$-chain

If $c$ is a singular $1$-cube in $\mathbb{R}^2 \backslash \left\{ (0,0)\right\}$ with $c(0)=c(1)$, show that there is an integer $n$ such that: $c-c_{(1,n)} = \partial (c^2)$ for some $2$-chain $c^2$.
JimJones
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