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
3
votes
0 answers

The set of critical value of polynomial functions

Let $f$ be polynomial function: $\mathbb{R^n} \rightarrow \mathbb{R}$. Show that the set of critical values of $f$ is finite.
user52523
  • 1,151
3
votes
3 answers

induced connection on open sets

Let $M$ be a smooth manifold. If $\nabla$ is a linear connection on $M$, I would like to induce a unique linear connection on an open subset $U\subseteq M$. I know that for all $p\in U$ there is a natural isomorphism $T_pU\cong T_pM$, so I can…
Dubious
  • 13,350
  • 12
  • 53
  • 142
3
votes
2 answers

Verification of Frenet Serret

I'm trying to show that (1) $T'\times T'' = k^2(kB +\tau T)$ $T' = \kappa N$, from Frenet Serret $T'' = \kappa'N + N'\kappa$, but the algebra didn't follow when I tried to substitute this on the Left hand side, of (1) above
Buddy Holly
  • 1,189
3
votes
1 answer

Show that the area of a regular tube of radius $r$ around a curve $\alpha = 2\pi r $(length of $\alpha$)

This is a question from do Carmo exercises, Sec 2-5. I know I just have to compute the area by using First Fundamental form, area $$ = \int\int\sqrt{EG-F^2}du dv$$ for a paramatrisation $x(u, v)$ of the tube surface, but I can't think of a…
john doe
  • 1,075
3
votes
2 answers

Parametric curve on cylinder surface

Let $r(t)=(x(t),y(t),z(t)),t\geq0$ be a parametric curve with $r(0)$ lies on cylinder surface $x^2+2y^2=C$. Let the tangent vector of $r$ is $r'(t)=\left( 2y(t)(z(t)-1), -x(t)(z(t)-1), x(t)y(t)\right)$. Would you help me to show that : (a) The…
tes
  • 395
3
votes
1 answer

Representing a unit speed curve on a sphere in terms of its Frenet Frame

Let $\alpha$ be a unit speed curve with positive curvature $\kappa \gt 0$ and non-zero torsion $\tau \ne 0$, lying on a sphere of radius $r$ centred at $c \in \Bbb{R}^3$. Show that $\alpha - c = -\frac{1}{\kappa}N -…
Helen Peters
  • 73
3
votes
1 answer

Covariance and contravariance

Given a vector space $V$, a vector $v \in V$ can be written in components with respect to different bases, say $X$ and $Y$. Now when i make a transformation from $X$ to $Y$, the components of the vector are transforming contravariantly. Now the dual…
kot
  • 313
3
votes
3 answers

What's the difference between $S^1, \mathbb{P}^1$?

I know that topologically, $S^1, \mathbb{P}^1$ is isomorphic. In doing differential geometry can I assume they are diffeomorphic as a smooth manifold? What is the difference between them?
Gobi
  • 7,458
3
votes
2 answers

If I am told to reparameterize a curve by arc-length, what is the curve originally parameterized by?

In differential geometry, you are often asked to reparameterize a curve using arc-length. I understand the process of how to do this, but I don't understand what we are reparameterizing from. What is the curve originally parameterized by (before we…
mmaclaren
  • 31
3
votes
0 answers

Curvature of a curve $\vec{r}(s)=(3+s)\hat{x}+\hat{y}$

I am having problem in calculating the curvature of the following curve.$$ \vec{r}(s)=(3+s)\hat{x}+\hat{y} $$ where $s$ is the arc length parameter. I know that $$…
I am pi
  • 617
3
votes
1 answer

Proving $\nabla F(x,y,z)$ is normal to the surface $F(x,y,z)=0$

What would be a simple way to prove that $\nabla F(x,y,z)$ is normal to the surface $F(x,y,z)=0$? I was wondering if anyone had a simple way to do this. Thanks in advance
Freeman
  • 5,399
3
votes
2 answers

Hodge-star operator and wedge product on Lie algebra valued forms

Let $M$ be a an oriented riemannian manifold. I have seen the following definition for the Hodge-star operator acting on a differential form. Starting with $\beta\in \Omega^p(M)$ we have $$\alpha \wedge \star \beta =…
physicsmajor1
  • 67
3
votes
0 answers

Questions on the $n$-transitivity of $\operatorname{Diff}(M)$.

Recently, I tried to understand how to proof the $n$-transitivity of $\operatorname{Diff}(M)$ acting on a smooth connected manifold $M$. I found a proof here $n$-transitivity of $\operatorname{Diff}(M)$ acting on a smooth manifold $M$ and another…
Sov
  • 1,048
3
votes
0 answers

Smoothness of tensors on reductive coset manifolds

This comes from O'Neill's Semi-Riemannian Geometry, in the proof of Proposition 11.22. Given a reductive coset manifold $M = G/H$ of a Lie Group $G$ with Lie subspace $m$, if you fix the differential of the projection map $d\pi$ to be an isometry,…
L N
  • 31
3
votes
2 answers

Pull-back of a differential : I get confused with the variables

I edited again my message with the remarks done in the comments. I have a 2-form : $$\alpha=\alpha_{\mu \nu} dx^\mu \wedge dx^\nu$$ I want to compute the pull back $F^{*}(d \alpha)$ to show that : $F^{*}(d \alpha)=dF^{*}( \alpha)$ But I make a…
StarBucK
  • 689
1 2 3
…
99 100