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
1 answer

Surface reconstruction from Laplace-Beltrami eigenfunctions

Consider a smooth, compact Riemannian surface $\mathcal{S}$ in $\mathbb{R}^3$ and suppose we are given the complete set of eigenfunctions $\{\phi_i\}$ of the associated Laplace-Beltrami operator. Is this information sufficient to fully reconstruct…
madison54
  • 3,077
3
votes
2 answers

Finding the metric of a surface embedded in $\mathbb{R}^3$

I have a problem about finding the metric of a surface defined by $x=\rho\cos\varphi,\ y=\rho\sin\varphi,\ z=\rho^2$, embedded into $\mathbb{R}^3$, where $ds^2=dx^2+dy^2+dz^2$. I have literally no idea how to do this. Worse perhaps, is I find the…
FireGarden
  • 5,835
3
votes
0 answers

Stereographic projection from sphere to $\mathbb{R}^2$

This question is from my tutorial problem set: One way to define a system of coordinates for the sphere $S^2$ given by $x^2+y^2+(z-1)^2=1$ is to consider the stereographic projection $\pi:S^2-\{N\} \to \mathbb{R}^2$ which carries a point $(x,y,z)$…
Alexy Vincenzo
  • 5,400
3
votes
2 answers

Show that $N$ is contained a plane.

Let $\alpha$ be a planar curve in $\mathbb R^3$, contained in plane $P$. Show that its normal vector $N$ at every point is in the plane P also. The only thing I know is that the torsion is 0, but I don't how to relate it to N. Please give me some…
JSCB
  • 13,456
  • 15
  • 59
  • 123
3
votes
3 answers

Prove that a manifold is not orientable

I have found a proposition who says: A manifold M is not orientable if it contains a Moebius band. How can I prove this?
andreasvr
  • 769
3
votes
2 answers

Intuition behind smooth functions.

Smooth functions $f(t)$ are those such that $\frac{d^nf(t)}{dt^n}$ exists for all $n\in\Bbb{N}$. I understand the intuition behind smoothness for functions like $f(t)=| t|$ and $f(t)=\sqrt{t}$. $f(t)$ has a "sharp" (and hence non-smooth) turn at…
user67803
3
votes
0 answers

Connection on a restricted bundle

Connection on a restricted bundle For a principal fiber bundle with a base $M$ and a structure group $G$ (for simplicity Lie group): $P(M,G)$ there is a connection form $\omega$. Is it true that if a fiber bundle restriction $P(M,G) \to Q(M,H)$…
user124443
  • 31
3
votes
2 answers

Equation of a plane from cross product

I'm working from Do Carmo, and I ran into another snag. More specifically, 1.4.5: Given points $p_1, p_2, p_3 \in \mathbb{R}^3$, show that the following expression gives the equation for the plane containing these points: $((p-p_1) \wedge (p-p_2))…
Lost
  • 2,095
3
votes
1 answer

Metric on tangent vectors to tangent space

Let $M$ be a Riemannian manifold and $p$ be a point of $M$. Let $v$, $v'$ be tangent vectors to $M$ at $p$. Of course we have $\langle v,v'\rangle_p$ defined. Let $u$, $w$ be tangent vectors to $T_p(M)$ at $v$. How is $\langle u,w\rangle_v$…
user120713
  • 31
3
votes
1 answer

Most natural symplectic structure?

Suppose I have 2-dimensional manifold embedded in $\mathbb{R}^3$. It's clear that the most natural Riemannian metric is the one induced by the usual inner product. What about symplectic forms? Is there a canonical symplectic form I can put on this…
user908123
  • 41
3
votes
1 answer

Differential Geometry review questions. Need help

I have a final coming up in Differential Geometry and we got a review worksheet and I am having serious trouble with two problems. I'm still chugging along at them but I need help understanding. I know we learned the 2nd problem (#5) while I was…
Metzky
  • 41
3
votes
0 answers

Definition of contact metric structure

I know this is a rather stupid question, but I still need to ask (and I am a physics student, so please excuse me using components): In Blair's book and many other litereatures, the definition of a contact metric structure ($\kappa, R, g, \Phi$) …
Lelouch
  • 637
3
votes
1 answer

is a plane smooth surface?

let f(u,v)=a + u.p + v.q : the equation of the plane where p,q are unit vectors perpendicular to each other. a a point on the plane. I do not understand how f can have partial derivatives of all orders, since derivative of wrt. u and v are p and q,…
104078
  • 1,919
3
votes
1 answer

Injectivity radius estmiates

Let $x\in M$ be a point, whose injectivity radius is $r_x$. So is it true that for any point $y \in B(x, r_x)$, the injectivity radius at $y$ is at least $r_x- d(x, y)$? is there any book has this result or if it is false what is the count example?
Sun
  • 1,018
3
votes
2 answers

Are critical points fixed?

Let $M$ be a smooth manifold (compact, connected, without boundary and oriented if you wish) with a smooth action of $S^1$. Let $f:M\rightarrow\mathbb{R}$ be an invariant function $f$. I know how to prove that a fixed point of the action is a…
A. Bellmunt
  • 1,582
  • 9
  • 15
1 2 3
…
99 100