Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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find the torsion and the curvature of this curve… (it's horrible)

Let's consider the following curve: $\varphi(t)=\begin{cases} (t,0,e^{-\frac{1}{t^{2}}}) & t>0\\ (0,0,0) & t=0\\ (t,e^{-\frac{1}{t^{2}}},0) & t<0 \end{cases} $ I have to compute ...
5
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1answer
90 views

Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$

How can I prove it? This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof. (Proving this will help me apply Stokes' theorem to ...
2
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1answer
143 views

What does it mean to “calculate in local coordinates” on a manifold?

In differential geometry textbook one sometimes reads "calculating in local coordinates, we obtain..." What does this expression mean? Say, $M$ is a smooth manifold and $h$ is a function on $M$; what ...
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1answer
350 views

Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms.

In problems 21 and 22, Rudin defines the differential forms $\eta=\dfrac{xdy-ydx}{x^2+y^2}$ and $\zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3}$ and the reader is asked to prove ...
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237 views

geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
2
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0answers
102 views

A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, has written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...
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1answer
169 views

General quasilinear PDE - derivation of characteristic equation

A general inhomogeneous quasilinear PDE is given as $a(x,t,u)u_t + b(x,t,u)u_x = c(x,t,u)$. In the derivation of the characteristic equations it says one can consider the solution to this PDE as the ...
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1answer
256 views

Torus with positive sectional curvature.

There was this question, whether a torus in dimension n, $T^n$, can carry a riemannian metric with positive sectional curvature. A read a proof, which goes as follows: $T^n$ is complete, because ...
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3answers
812 views

Can the curl operator be generalized to non-3D?

In three dimensions, the curl operator $\newcommand{curl}{\operatorname{curl}}\curl = \vec\nabla\times$ fulfils the equations $$\curl^2 = ...
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110 views

Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
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1answer
43 views

How to determine if a surface exists

Given that the coefficients of the first fundamental form are $E=G=1\ F=0$ and the coefficients of the second fundamental form are $L=1\ M=0\ N=-1$. How does one determine if the surface exists?
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59 views

deRham cohomology of a manifold with covering space $S^{n}$

Let $\pi: S^{n}\rightarrow M$, $n>1$ be a covering map, $M$ being an orientable manifold. Show that $H^{k}_{deR}(M)=0$ for $1\leq k<n$. I know how to do for $H^{1}_{deR}$, but my argument fails ...
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1answer
258 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
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0answers
229 views

do Carmo: near isolated zeros, killing field tangent to geodesic spheres

Exercise 3.5b of do Carmo's Riemannian Geometry asks the reader to prove that given a Killing field $X$ on a manifold $M$, an isolated zero $p$ of $X$, and a normal neighborhood $U$ of $p$ in which ...
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1answer
99 views

Conformal maps question

Let $\psi:\Bbb R^2\to \Bbb R^2$ be given by $\psi(x,y)=(u(x,y),v(x,y))$ where $u$ and $v$ are differentiable and satisfy $u_x=v_y, u_y=-v_x$. Prove that $\psi$ is a local conformal map from $\Bbb ...
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82 views

Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$

Let $M$ be a smooth real manifold. I want to show that we have an isomorphism of real vector space $\Gamma(TM)$ of all smooth sections of $TM$ (i.e. of vector fields on $M$) and of real vector space ...
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1answer
165 views

question about Gaussian map

I have questions. Can anyone help me to get the idea or figure out this problem. compute the Gaussian and mean curvature for torus. notice the metric for torus is X(U,V)=((a+b cos(u))cos(v),(a+b ...
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2answers
616 views

How to quantify the distance between matrices with an irrelevant rotation factor?

Suppose you have two invertible matrices $A$, $B$ in $\mathbb{R}^{n\times n}$, that is, $A,B\in GL(n)$. You want to define a distance between them that ignores arbitrary rotational factors, so ...
3
votes
2answers
142 views

Is $\omega_n$ exact in $\mathbb R^n -\{0 \}$?

For $n \ge 2$ consider the differential form $\omega_n=r^{-n} \sum_{i=1}^n(-1)^{i-1}x_idx_1 \wedge \ldots \wedge dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_n$, defined on $\mathbb R^n \setminus ...
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1answer
127 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
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1answer
107 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
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1answer
106 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
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2answers
265 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
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1answer
347 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
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1answer
288 views

How can a $C^1$-continuous surface have infinite curvature?

Short version: Apparently it is possible for a $C^1$-continuous parametric surface to (locally) have infinite (Gaussian) curvature. I find this quite counter-intuitive, because I always thought that ...
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1answer
98 views

Surfaces without conjugate points

I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
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0answers
96 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
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1answer
109 views

If $ \eta $ and $ \varphi $ are closed differential forms, then prove that $ \varphi \wedge \eta $ is a closed differential form.

Let’s assume that $ \eta $ and $ \varphi $ are closed differential forms. Then how can I prove that $ \varphi \wedge \eta $ is a closed differential form as well? Please explain how to solve this ...
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3answers
364 views

How to differentiate a differential form?

Please explain me the idea of differentiating differential forms (tensors). Example: compute d(xdy + ydx) The answer is known, we should have 0. What's the rule?
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votes
1answer
117 views

linear equivalence of divisors given by sections of a linebundle

The topic of divisors is new to me and I wonder if I understand a few things correctly. In my situation I have an analytical Manifold $M$ of dimension $2$ and its compactification $\overline M$. I ...
4
votes
1answer
280 views

Do Carmo: Linear Killing field anti-symmetric?

In Exercise 3.5a of Riemannian Geometry, do Carmo defines a vector field $v$ on $\mathbb{R}^n$ to be linear if it's linear as a map $v\colon \mathbb{R}^n \to\mathbb{R}^n$. He then asks the reader to ...
9
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1answer
130 views

Characterizing singularities using sheaves of smooth functions

Short version: Let $H\subset M$ be a closed subset of a smooth manifold. Equip $H$ with the sheaf $\mathcal{F}$ of smooth functions (so that a section over an open $U$ is the restriction to $U$ of ...
2
votes
1answer
286 views

Solve the Poisson Equation on a Riemannian Manifold

Imagine that I have a field that obeys to the Poisson equation. To solve the equation, in my bag of tools I only have the divergence theorem or the Fourier/Laplace transform. They usually are enough ...
6
votes
1answer
200 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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0answers
87 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
4
votes
2answers
265 views

Property of normal coordinates

Let $M$ be a Riemannian manifold and $\nabla$ the Levi-Civita conection. I need to prove the following. Let $B$ be an open ball of radius $r$ in $T_pM$ such that $\left.exp_p\right|_B$ be a ...
2
votes
1answer
94 views

Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?

$r=\sqrt{x^2+y^2+z^2}$ of course. If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$ I've managed to figure out 2 ...
5
votes
1answer
196 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
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0answers
187 views

Difference between distance between two points and metric

if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
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1answer
54 views

Showing a Frame is Orthonormal

I know this is really simple. But I don't know how to carry out the calculation, I can only "see" why the following is a orthonormal frame. Let $$E_1 = \frac{x}{r}\frac{\partial }{\partial x} + ...
0
votes
1answer
65 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
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votes
2answers
2k views

Left-Invariant Vector Field of a Lie Group

How do I tell if a vector field on a Lie Group is left-invariant? I have the technical definition. But, I want to understand given a specific vector field what should I do to test if it is ...
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1answer
142 views

Smoothness Criterion for Vector Fields

I'm going to just write the proof (straight from Lee), my question is about the $(*)$ stared part. Let $M$ be a smooth manifold and let $X:M \to TM$ be a (rough) vector field. If $(U,(x^i))$ is any ...
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2answers
142 views

what is $C^{-\infty}(\mathbb{R})$

Thanks in advance. what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way ...
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1answer
38 views

Closure of a set's cone

working in $\mathbb{R}^3$ , say I'm looking at the set : $\{1\} \times S^1$ denote E for the cone of the set above, is E a closed set? (I think it is) if not, what is it's closure? thanks.
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34 views

How is integration of differential form defined as, and how to calculate it

How is integration of differential form defined as? And how does one calculate the value of integration?
4
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1answer
223 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
7
votes
2answers
677 views

How to embed Klein Bottle into $R^4$

I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem. The problem is: Show that the mapping $G:\mathbb{R}^2\to\mathbb{R}^4$ given by $$G(x,y)=((r\cos y+a)\cos x,(r\cos ...
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2answers
71 views

Pullback map and its equality consequence

On my textbook, it says: $$F^{\star}(dy^i) = \sum_{j=1}^{n} \frac{\partial y_i}{dx_j}dx_j$$ where $F^{\star}$ is a pullback map, map $F: M_1 \rightarrow M_2$ and $dx_j$s are forms on $M_1$ and ...
5
votes
2answers
69 views

Tangent Space of $\operatorname{Aut}(T_e G)$

Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.