1
vote
0answers
20 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
37 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 $h = 1, \lambda = \frac{m}{n}$, ...
0
votes
0answers
10 views

Geodesic trajectories of 3D hyperbola

Consider a 3-dimensional space given by the set of points {(x,y,z),x∈R,y∈R,z>0} with the metric ds2=a/z2(dx2+dy2+dz2). b) Consider two geodesic trajectories with initial conditions ...
0
votes
0answers
21 views

for what values of c is f a regular surface

I just want to make sure I understand what's going on here (from do carmo's differential geometry book): (a)$f(x,y,z) = (x + y + z - 1)^2$ (b)$f(x,y,z) = xyz^2$ For each function, find the critical ...
1
vote
0answers
40 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
22 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
43 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
52 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
38 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
53 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
81 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
50 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
33 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
49 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
61 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
44 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
53 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
25 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
22 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
40 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
106 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
51 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
38 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
42 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
32 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
68 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
68 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
46 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
50 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 ...
1
vote
1answer
76 views

An basic question of linear algebra and the vectorial spaces.

I'm studying Differencial Geometry for a exam and I'm having some trouble to do this. In this course we viewed dual spaces, forms, Gauss Bonnet Theorem... Hm.. this is it! Let $V$ a vectorial ...
0
votes
1answer
58 views

A basic application of the Gauss-Bonnet theorem for polygons.

I'm studying for an exam and having trouble to apply this theorem. What the exercise says... Let the regular geodesic polygon $P_n(x)$ with $n$ sides and center on the Poincaré's Disc, where $x$ ...
2
votes
0answers
54 views

Differential Geometry in R^3. Show there exists a unique unit speed circle given a unit speed curve.

Here is(are) the question(s). Let $f(t): (-t_0-\epsilon, t_0 +\epsilon)\to \Bbb R^3$ be a unit speed parametrized curve in $\Bbb R^3$. Suppose that $k(f(t_0)) > 0$, where $k$ is the curvature ...
1
vote
1answer
84 views

$V$ vector field, $\omega$ one-form, $V(\omega(V))$=?

(1-forms) Let $X$ be a manifold and $\omega \in \Omega^1(X)$ be a smooth 1-form, and $V, W \in V^{\infty}(X)$ smooth vector fields on $X$. Then, $\omega(V ), \omega(W ) \in C^{\infty}(X)$ are ...
0
votes
2answers
118 views

curvature of helix

Here is the curve of a helix parametrized by its arc length $\alpha(s) = ( a\cos(\frac{s}{c}), a\sin(\frac{s}{c}), b(\frac{s}{c}) ), s \in \mathbb{R}$ such that a$^2$ + b$^2$ = c$^2$. The curvature ...
1
vote
1answer
90 views

Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d $ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)} $ be a family in $\mathrm{Diff}(\Omega)$ such that $ \phi_0 = id_{\Omega}$ and ...
0
votes
1answer
63 views

Why does a singular point create a cusp or a node on the trace?

What the geometrical meaning for a singular point of a parametric curve? i.e Suppose $\alpha$(t) = (x(t), y(t)), then $\alpha '$ (t) = (x'(t), y'(t)). A singular point at t$_0$ is when $\alpha ...
0
votes
0answers
35 views

Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
1
vote
2answers
58 views

Four circles & a square in a circle

Radius of the big triangle is $2$. ABCD is a square. What is the difference between $T_{1}$ and $(M_{1}+M_{2})$. I have solved it already though I don't know if my answer is right or wrong. My ...
2
votes
2answers
80 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
4
votes
1answer
190 views

Christoffel Symbols Equality Solution?

They changed the exercise, so I tried to solve it again: I have to prove the following: Let $\Omega \subseteq \mathbb{R}^d$ be open and $g$ a metric field on $\Omega$. For every $\phi \in ...
4
votes
3answers
805 views

Christoffel symbols equality

I have to prove the following: Let $\Omega \subseteq \mathbb{R}^d$ be open and $g$ a metric field on $\Omega$. For every $\phi \in \mathrm{Diff}(\Omega)$ let $\Xi^i_{jk}[\phi]$ be functions on ...
1
vote
2answers
105 views

Question about differential form

$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!
2
votes
0answers
82 views

A formula for the holomorphic sectional curvature.

I tried to compute the holomorphic sectional curvature of a hypersurface of ($\mathbb{C}^{n+1}$, std metric, i), but I failed. $$ V_{k}=\{(z_{0},...,z_{n})\in \mathbb{C}^{n+1} | \sum_{j}z_{j}^{k}=0\}- ...
1
vote
1answer
33 views

Show that the geodesics in normal coordinates have unit speed

Given the normal coordinate system $y^ \alpha$, show that the geodesics are radial lines of the form $tv$, where $v$ is a vector of length 1. I have managed to show that the geodesics are radial ...
1
vote
1answer
67 views

If there exists a global section then the principal bundle is trivial - problem with smoothness

Let $\pi \colon P \to B$ be a principal $G$-bundle and let $s \colon B \to P$ be it's smooth section. In order to show that $P \simeq B \times G$ I define the map $\varphi \colon P \to B \times G$ by ...
0
votes
3answers
36 views

Proving an identity related to the torsion of a connection.

Let $\nabla$ be a connection, and let $T(X,Y) = \nabla_{X}Y - \nabla_{Y}X - [X,Y]$ be the torsion of $\nabla$. I am trying to prove that if $f$ is a smooth function, then $fT(X,Y) = T(fX,Y)$. Using ...
2
votes
1answer
30 views

Compact hypersurface of $\mathbb{R}^{n+1}$ with positive curvature is diffeomorphic to $S^n$.

I have a compact hypersurface $M$ of $\mathbb{R}^{n+1}$ with positive curvature. I need to show that it is diffeomorphic to $S^n$. The hint is to consider the shape operator $A_{\nu_p} x$, where ...