1
vote
1answer
68 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
0
votes
0answers
26 views

Showing that the two-sheeted cone is is not a regular surface

I was given the following problem in class: Show that the two-sheeted cone, with its vertex at the origin, that is, the set $$\{(x, y, z) \in \mathbb{R^3}\,|\, x^2 + y^2 - z^2 = 0\}\text{,}$$ ...
1
vote
1answer
48 views

Visualizing the level sets for this function

Let $F: T^{2} \to \mathbb{R}$ be given by $(x_{1}, x_{2}, x_{3}) \mapsto x_{2}$. Recall that $$T^{2} = \{(x_{1}, x_{2}, x_{3}) : \left(\sqrt{x_{1}^{2} + x_{2}^{2}} - R \right)^{2} + x_{3}^{2} = ...
0
votes
1answer
17 views

Verification on Intersection of Tangent Lines.

Given a function $\gamma (t) = (-1,4)(2-3t)^2 + (1,0)(3t - 1)^2 $ figure out the control point $P1$ which exist at the intersection of the tangent lines of $P0$ = $\gamma (0) $ and $P2$ = $\gamma (1) ...
0
votes
0answers
22 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
0
votes
1answer
55 views

Question about Alternating forms

So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
2
votes
0answers
38 views

Verifying the Divergence Theorem for Half of a Sphere

Here is an exercise that I was assigned for homework: .......................................................... To the bottom left, I have scanned an example problem for verifying the divergence ...
0
votes
0answers
10 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
0
votes
0answers
20 views

Determining an arclength and a unit vector - asking for corrections.

I'd like you to help me answer/solve the two problems. There are also my attempts. I'd be appreciated if you could correct them and help me doing the right way. Let $\sigma:U\to\mathbb{R}^{3}$ be a ...
2
votes
1answer
48 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
1
vote
0answers
42 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
0
votes
1answer
53 views

Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
1
vote
1answer
45 views

Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
1
vote
1answer
43 views

Gaussian curvature of a parallel surface

Question 11 section 3.5 in Do carmo part c.Let a surfae x have constant mean curvature equal to c does not equal 0 and consider the parallel surface to x at a distance 1/2c. Prove that this parallel ...
0
votes
1answer
18 views

Mean curvature an asymptotic directions

I'm a little confused by this question I've come across in do carmo while studying for my final. (Sect 3.2 #7) Show that if the mean curvature is zero at a nonplanar point, then this point has two ...
0
votes
1answer
36 views

How to show left-invariant frame?

I am working on the following problem and I'm stumped. What do I need to show? What does "left-invariant frame" mean? Consider $\mathbb{S}^3$ as the unit sphere in $\mathbb{R}^4$ with coordinates ...
1
vote
2answers
37 views

How to show the symplectic group is a submanifold of $GL(n,\mathbb{H})$?

I am trying to show that the symplectic group $Sp(n) =\{A\in GL(n,\mathbb{H})\mid \overline{A}^TA=I\}$ is a regular submanifold of $GL(n,\mathbb{H})$ but I am stuck. Any help would be appreciated.
1
vote
3answers
110 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
2
votes
3answers
121 views

Manifolds and their dimension

Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on $\mathbb R^3$. I have difficulty in ...
1
vote
1answer
41 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
1
vote
0answers
28 views

Show an immersion is locally one to one using the inverse function theorem

Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint?
1
vote
0answers
45 views

Geometry of Curves and Surfaces

The following is a question regarding Elementary Geometry of Differentiable Curves. Venture a guess of the number of irregular values of $t$ for the trochoid with $h = 1, \lambda = \frac{m}{n}$, ...
1
vote
0answers
45 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
0
votes
1answer
33 views

locate critical points/values and show where function is a regular surface

Let $f(x,y,z) = (x + y + z - 1)^2$. a. Locate the critical points and critical values of $f$. b. For what values of $c$ is the set $f(x,y,z) = c$ a regular surface? a. So, to locate critical points ...
1
vote
2answers
54 views

Solving a certain differential equation when assuming a surface of revolution is minimal

The problem is the following: Consider the surface of revolution $$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$ If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ ...
0
votes
1answer
117 views

Asymptotic Curves and Lines of Curvature of Helicoid

I have a question that asks me to find the asymptotic curves and lines of curvature of the helicoid given by: x = v Cos[u], y = v Sin[u], z = c*u, for some fixed real c. Can you show me how best to do ...
0
votes
0answers
49 views

Prove that the tangent space TpM is vector space .

We have to prove that $(T_p M , + , *)$ is vector space " + " : $T_p M \times T_p M \to T_p M$ $(X_p + Y_p)(f) = X_p(f)+Y_p(f)$ So we have to prove that $(T_p M ,+)$ is abelian group : ,, + " is ...
5
votes
1answer
66 views

Problem about parallel curve- differential geometry

Let $\alpha (s)$ , $s\in [0,L]$, be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve $\beta(s)=\alpha (s) +\lambda n(s)$, where $\lambda$ is a positive ...
8
votes
1answer
92 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
0
votes
2answers
51 views

How to identify a curve

Suppose that a curve $\mathbf\gamma$ in $\mathbb R^3$ has constant strictly positive curvature function $\mathbf\kappa(s)$, and constant non-zero torsion function $\mathbf\tau(s)$. Prove that the ...
0
votes
1answer
35 views

parameterized ellipse, error in proof of a theorem?

A question from the book "Elementary Differential Geometry" from A Pressley Consider the ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$, where $p>q>0$ The eccentricity of the ellipse is ...
1
vote
0answers
61 views

tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
3
votes
1answer
63 views

An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
2
votes
1answer
55 views

Meridians of surfaces of revolutions

First off, I know there is another question asking the same thing, but that one was concerning where to start, whereas for this one, I'm almost complete, but I can't get something at the end to ...
2
votes
1answer
57 views

Calculate the integral of a 2 form

I am trying to compute the integral $$ \int\int_{S}\frac{1}{x}dy\wedge dz+\frac{1}{y}dz\wedge dx+\frac{1}{z}dx\wedge dy $$ over an ellipsoid given by $$ ...
2
votes
0answers
26 views

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 ...
2
votes
1answer
24 views

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 ...
0
votes
1answer
41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
0
votes
1answer
42 views

Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
3
votes
1answer
120 views

Curvature of a curve lying on a sphere?

This is a sample question from a multivariate calculus class. Any insight would be appreciated. Suppose the curve $\mathbf{r} = \mathbf{r}(s)$ is parametrized by a natural parameter and lies on the ...
1
vote
0answers
52 views

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 ...
0
votes
1answer
53 views

Finding the tangent plane of a point of a curve when using implicit differentiation

I need to find the tangent plane of this surface: $$(z-1)^3=\sin(y^2)e^{xz}$$ at the point $(0, \sqrt \pi ,1)$ I find $dz \over dx$ and $dz \over dy$ $${dz \over dx}={-e^{xz}\sin(y^2)z \over ...
2
votes
1answer
49 views

Parallel Curve of Regular Plane Curves

Let $\gamma$ be a regular plane curve and let $\lambda$ be a constant. The parallel curve $\gamma^\lambda$ of $\gamma$ is defined as $$\gamma^\lambda(t)=\gamma(t)+(\lambda)n_s(t),$$ where $n_s$ is the ...
0
votes
0answers
42 views

Given the normal vector n(s), determines the curvature k(s) and the torsion

Given the normal vector n(s) of a curve $\alpha$, with non zero torsion everywhere, determines the curvature k(s) and the torsion $\tau$(s) of $\alpha$. I am first trying to show the following which ...
1
vote
1answer
42 views

Line containing a vector equation

In Do Carmo 1.5.1d, it asks: Show that, for the parametrized curve $\alpha(s) = (a$ cos $\frac{s}{c}, a$ sin $\frac{s}{c}, b\frac{s}{c})$, the lines containing the normal $n(s)$ passing through ...
3
votes
2answers
71 views

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 ...
0
votes
1answer
35 views

Finding limit of a parametrized curve

Related to this question: Parametrized curve tangent to a line I'm working on Do Carmo 1.3.5c, which is: Given a parametrized curve $\alpha(t) = (\frac{3at}{1+t^3},\frac{3at^2}{1+t^3})$ and the ...
3
votes
1answer
74 views

Does there exist a surface homemomorphic to a torus with positive Gaussian curvature?

This is a problem from the my last exam in Differential Geometry II and I didn't solve it. I'm studying again, but without success. So I need help. Does there exist a surface $S \subset ...
3
votes
1answer
47 views

How to verify that this is a submanifold

Let $ g: \mathbb{R}^2 \to \mathbb{R}^2 $ , $ g (x, y) = (x^2-y^2, y) $ be a differentiable map. Let $ r $ the line passing through $(1, 0) $ parallel to the $ y-$axis. Prove that $ g^{-1}(r) $ is a ...
0
votes
1answer
57 views

Parametrized curve tangent to a line

I'm working from Do Carmo's book Differential Geometry, and I was a bit confused about one question - 1.3.5 in particular. The question asks: Let $\alpha:(-1,\infty) \to \mathbb{R}^2$ be a ...