Mathematics Stack Exchange
Mathematics Stack Exchange

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
5
votes
2 answers

Covariant Derivative of a vector field - Parallel Vector Field

I'm having trouble to understand the concept of Covariant Derivative of a vector field. The definition from doCarmo's book states that the Covariant Derivative $(\frac{Dw}{dt})(t), t \in I$ is defined as the orthogonal projection of $\frac{dw}{dt}$…
cryptow
  • 479
5
votes
2 answers

Covariant derivative in cylindrical coordinates

I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57). Equation (48) shows that the covariant derivative is: $$A_{j;k} =…
OSE
  • 235
5
votes
2 answers

existence of hemisphere

If $ \beta: (a,b)\rightarrow \mathbb{S}^2 $ is a simple closed curve such that $ \int_a^b\Vert\beta^{\prime}(t) \Vert dt<2\pi$ then there is an open hemisphere (or any rotation of this) containing the image of $\beta $ thanks!
helmonio
  • 725
5
votes
1 answer

Curvature and torsion changes related to Frenet frame choice

Let $\gamma(s)$ be a unit-speed curve in $\mathbb{R}^3$. Let $t = \dot{\gamma}(s)$, $n = \frac{\dot{t}}{\left \| \dot{t} \right \|}$ and $b= t \times n$. The vectors $(t,n,b)$ form what is called a Frenet frame in a point $s$. Define the curvature…
user14174
5
votes
1 answer

parallel curvature imply constant Ricci and scalar curvature

$\text{Suppose we have} \nabla R = 0 $, where R represents curvature tensor, Prove that Ricci curvature and scalar curvature are constant.
Chen Jie
  • 103
5
votes
1 answer

$[D,D']$ where $D$ is a derivation and $D'$ is skew

This is a proposition in 33 page of Foundation in Differential Geometry - KN I need some detail. Let $D^r(M)$ be a set of $r$-form. Then derivation (resp. skew-derivation) of degree $k$ is a linear mapping from $D^r(M)$ to $D^{r+k}(M)$ s.t. $$…
HK Lee
  • 19,964
5
votes
1 answer

Gauss curvature using metric and Riemannian curvature

I learnt that the Gauss curvature is given by: $$K = \frac {eg - f^2}{EG - F^2}$$ where $E, F, G$ are coefficients of the first fundamental form and $e,f, g$ are coefficients of the second fundamental form. However, in a proof that I am reading, I…
user145817
  • 73
5
votes
1 answer

Various Parallels on a Torus

Consider the torus of revolution generated be rotating the circle $\{(x,y,z) \in \mathbb{R}^{3}: (x − a)^{2} + z^{2} = r^{2}, y = 0$ }, where $a > r > 0$, around the $z$-axis. The parallels generated by the points $(a + r, 0)$, $(a − r, 0)$, $(a,…
kevin
  • 561
5
votes
2 answers

Gaussian curvature $K$ of of orthogonal parametrization $X$

Let $X$ be an orthogonal parametrization of some surface $S$. Prove that the Gaussian curvature $K = - \frac{1}{2 \sqrt{E G}} ((\frac{E_{v}}{\sqrt{E G}})_{v} + (\frac{G_{u}}{\sqrt{E G}})_{u})$, where subscripts denote partial differentiation of the…
kevin
  • 561
5
votes
1 answer

Define curvature and curvature of a circle

Question:(a) Define the curvature function $\kappa$ of a plane curve. The curvature of $\kappa$ of a plane curve is the amount of turning in the osculating plane. In other words it decribes the speed of rotation. Is how I defined it okay? I feel…
Ruth Gutierrez
  • 883
  • 2
  • 13
  • 25
5
votes
1 answer

constant speed of curve,regular curve, and reparametrization by an arc length

Question: Show that the curve $\alpha(t)=(sint,t,-cost)$ has a constant speed. Is this curve regular curve? Then find a reparametrization of this curve by an arc lenth. The curve of $\alpha$ has a constant speed and is a regular curve that can be…
Ruth Gutierrez
  • 883
  • 2
  • 13
  • 25
5
votes
1 answer

Flow on manifolds and Lie bracket.

I'm currently reading through some notes on Lie Theory online, and I've stumbled across the following question, which I'm totally stumped by. "Let X,Y be two commuting complete vector fields on a manifold M, that is $[X,Y]=0$. Show that the vector…
Shaf_math
  • 275
5
votes
0 answers

Is every second-order elliptic operator the Laplace-Beltrami operator of some metric?

Suppose I have a general elliptic operator $$Lu = \nabla \cdot M\nabla u$$ on $\mathbb{R}^2$, where $M(x,y)$ is a symmetric positive-definite $2\times 2$ matrix at every point $(x,y)$ and varies sufficiently smoothly over the plane. Does there…
user7530
  • 49,280
5
votes
2 answers

Parallel transport curve with corners

I am trying to understand the parallel transport along the triangle looking path on the sphere, as in http://en.m.wikipedia.org/wiki/File:Parallel_transport.png. The thing that confuses me is that the curve looks only piecewise continuous, so the…
Emil
  • 709
5
votes
1 answer

Show the cylinder is a regular surface

Show that the cylinder $(x,y,z) \in R^3; x^2+y^2=1 $is a regular surface and find parameterizations whose coordinate neighborhoods cover it. I'm going to be honest I saw this answer but I don't quite understand it. I am familiar with the…
Ruth Gutierrez
  • 883
  • 2
  • 13
  • 25
1 2 3
…
99 100