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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783 views

Why is Cartan Formula just an avatar of Leibniz rule?

In this video, Arnold says that the Cartan formula $$ \mathscr L_{\mathrm X} = d i_{\mathrm X} + i_{\mathrm X} d$$ is just an avatar of $(fg)' = fg' + f'g$. Why ?
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3answers
750 views

Why is mean curvature extrinsic?

I believe that an intrinsic property is dependent on a surface itself (not on how it is parameterized), and an extrinsic property may vary depending on the parameterization. Why is mean curvature ...
1
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1answer
127 views

derivative of the integral over a sphere of variable radius [Reference needed]

On a Riemannian manifold $(M,g)$, let $F(s)=\int_{\partial B_s(x_0)}udS$ where $u$ is a smooth function, ${\partial B_s(x_0)}$ is the (geodesical) sphere of center $x_0$ and radius $r$ dS is the ...
4
votes
1answer
139 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
3
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0answers
80 views

State of the art of the Implicit Function Theorem

What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ...
2
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0answers
296 views

Relationship between Gaussian, Normal and Geodesic Curvatures

How do I show that the square of the gaussian curvature is the sum of the squares of the normal and geodesic curvatures other than the one shown in page 38 of ...
0
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2answers
263 views

Question about unit normal of a sphere

I have question, I want some one explain how to compute the unit normal of sphere notice, gives $N=\displaystyle \pm \frac{1}{r}(h-a)$ where $a$ is center point of sphere $h(t)$ is unit speed of ...
4
votes
1answer
224 views

Trivial Tangent and Cotangent Bundles

If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?
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2answers
210 views

Ways to think about vector bundle

I'm studying manifold theory and I've got to the point of discussing the definition of a vector bundle. The definition is quite long and a bit confusing and I was wondering if someone with a bit more ...
0
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2answers
64 views

Find $T$, $N$, and $k$ for vector

$$ x(t) = ( t , \sin(4t), \cos(4t) )$$ I am unsure on what they are referring to here. would $T$ be the tangent and therefore: $$ x'(t) = ( 1 , 4\cos(4t) , 4\sin(4t) ) ?$$ Thanks :) Also, ...
3
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2answers
491 views

Difference between tangent space and tangent plane

I’ve avoided doing any manifold (regretting it somewhat) courses, however do have some understanding. Let $p$ be a point on a surface $S:U\to \Bbb{R}^3$, we define: The tangent space to $S$ at $p$, ...
2
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1answer
265 views

How to show that this set isn't a regular surface?

I'm trying to solve this exercise from Do Carmo's Differential Geometry of Curves and Surfaces, and I want a hint on how to do it. The exercise is: Is the set $S =\left\{(x,y,z)\in \mathbb{R}^3 \mid ...
3
votes
1answer
130 views

Zeros of the second fundamental form

Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically ...
2
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0answers
68 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
4
votes
3answers
291 views

Existence of continuous path connecting points on a plane

I've been thinking over this problem this weekend and although my investigations on it has led to other interesting theorems I am still nowhere close to solving it. For any set of disjoint ...
2
votes
0answers
65 views

an example of a curve such that…

Given differentiable functions $k(s),\tau(s)$ with domain $s\in (a,b)$, there exist a differentiable function $\gamma:(a,b)\to \mathbb R^3$ such that $k,\tau$ are it's respectively curvature and ...
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2answers
61 views

Could be possible that $\langle\gamma' , \gamma''\rangle$ is constant but $\|\gamma'(t)\|$ is not constant?

Let $\gamma: (a,b)\to \mathbb R^3$ be a differentiable curve. Note that if $\|\gamma'(t)\|=k\ne 0$ where $k$ is a constant, then $\|\gamma'(t)\|^2=\langle\gamma(t),\gamma(t)\rangle=k^2$. If we ...
3
votes
2answers
137 views

What is wrong with my proof? (A problem of tangent bundle)

I am proving $TS^1$ is diffeomorphic to $S^1\times\mathbb{R}$. The following is my proof and I think it is wrong, because I only use the fact that $S^1$ is 1-dimentional. However, I do not know how to ...
0
votes
1answer
50 views

Plan curve with zero area has at least two points of zero curvature

Let $\alpha=(x,y)$ be a smooth closed plan curve defined on $[a,b]\subset \mathbb{R}$. We can define the oriented area of $\alpha$ by $A=\int_{a}^{b} x(s)y'(s)ds$. So, if A=0 then there exists $t_1 ...
4
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0answers
323 views

Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
0
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1answer
227 views

Winding number of vector field on surface

I found a term "winding number of vector field with respect to another vector field" in a paper without definition. Because my paper I am reading is talking about the surface, so I don't know if I can ...
4
votes
1answer
197 views

Elliptic equation on riemannian manifolds

Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth ...
9
votes
1answer
338 views

Is the Structure Group of a Fibre Bundle Well-Defined?

Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
0
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1answer
58 views

Torsionfree connection on quaternionic manifold

Does the torsionfree connection (different from Levi Civita) always exist on quaternionic manifold? Where can I find more information about it, something from the very begining? For example, how it ...
4
votes
2answers
328 views

Bundle Automorphisms, Structure Groups and Gauge Groups

I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true. If $E$ is a $G$-principle bundle over $M$ then ...
0
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2answers
325 views

question about geodesic curvature of geometry

I have questions. Can anyone help me to get the idea or figure out this problem. What the formula of geodesic curvature and what is the easy formula to compute geodesic curvature for any surface. ...
4
votes
1answer
324 views

Why do we need Lie derivative?

If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative? In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
4
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1answer
260 views

Different definitions of handle attachment

This is an extremely technical question about handle attachments.  I am asking why two definitions are equivalent.  My question appears in the second to last paragraph after I've described the two ...
3
votes
1answer
204 views

What happens to the Frenet-Serret frame when $\kappa=0$?

I was considering the following question for 3D curves: Does zero curvature imply zero torsion? I think it's reasonable, because zero curvature implies the curve is a straight line, which lies in a ...
4
votes
1answer
255 views

A question about concept of pushforward

In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...
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vote
1answer
482 views

Proof that cone not diffeomorphic to plane

What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$? After some comments, I realize that this question The ...
0
votes
2answers
46 views

Why is an open disk not open when thought of as a subset of a plane in $E^3$?

An open disk is open in $E^3$. I believe this makes sense since the boundary of the open disk is not included with the set of interior points. However, apparently, an open disk is not open when ...
1
vote
0answers
120 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
4
votes
2answers
193 views

Closed ball not a manifold.

My book on differential geometry claims that a closed ball in $\Bbb R^n$ can never be a differentiable manifold because of the boundary points. The book doesn't really give an explanation for why this ...
5
votes
4answers
286 views

why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
3
votes
2answers
65 views

Action of $\text{SL}_2\mathbb{C}$ on $\mathbb{C}^3$ induces a 2:1 covering $\text{SL}_2\mathbb{C}\to \text{SO}_3\mathbb{C}$

Exercise 7.17 in Fulton's Representation Theory reads, Identify $\mathbb{C}^3$ with the space of traceless matrices in $M_2\mathbb{C}$ so that $g\in \text{SL}_2\mathbb{C}$ acts by $$A\mapsto ...
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2answers
208 views

Some topics in Differential Geometry for a beginner

So our instructor wants us to write a 3 page report on some fact in Differential Geometry relating to curves or surfaces. It can be a theorem, a fact or some special case. The only restriction is ...
4
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0answers
172 views

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
votes
1answer
78 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 ...
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1answer
118 views

How to calculate in local coordinates? [closed]

If $M$ is a smooth manifols what would be calculate a function $h$ defined on $M$ in local coordinates?
4
votes
1answer
325 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 ...
4
votes
0answers
224 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
votes
0answers
97 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 ...
2
votes
1answer
165 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 ...
1
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1answer
236 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 ...
11
votes
3answers
743 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 = ...
3
votes
0answers
104 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 ...
1
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1answer
42 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?
2
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0answers
58 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
216 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 ...