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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What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
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8 views

Fiber product of Fréchet manifolds

Let $X$, $Y$ be two manifolds modelled on Fréchet spaces. Let $f: X \longrightarrow M$, $g: Y \longrightarrow M$ be two smooth maps to a finite-dimensional manifold. Does the fiber product $X ...
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56 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
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1answer
40 views

Constructing lagrangian submanifold of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. To keep it simple, let us take $M = \mathbb{R}^{2n}$ with linear coordinates $(x^1,\ldots,x^n,y^1,\ldots,y^n)$ and the standard symplectic form $\omega = ...
3
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1answer
26 views

Calculating the pullback of a $2$-form

I have a $2$-form given by $\omega = dx \wedge dp + dy \wedge dq$ and a map $i : (u,v) \mapsto (u,v,f_u,-f_v)$ for a general smooth map $f : (u,v) \mapsto f(u,v)$. I want to calculate the pullback of ...
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1answer
42 views

Degree and Lefschetz number of a function

I'd like to check if I got the following computation right. Let $f:RP^3\rightarrow RP^3$ be given by $[x_0:x_1:x_2:x_3] \mapsto [x_0^2:x_1^2:x_2^2:x_3^2]$ I would like to compute the degree and the ...
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1answer
34 views

If a plane intersects a regular surface at exactly one point, then it is the tangent plane

Question Let a regular surface, $S$, intersect a plane, $P$, at only one point, $p_0 = (x_0, y_0, z_0)$ in $\mathbb{R}^3$. Show that the plane coincides with the tangent plane to the surface at ...
4
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1answer
40 views

connection laplacian on general vector bundles

As the title says, my question is about how to define the connection laplacian on general vector bundles. I think I understand how to define the connection laplacian on the tensorbundles: Let $M$ ...
0
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2answers
236 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. ...
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2answers
24 views

Inward and outward pointing tangent vectors?

If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart ...
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2answers
51 views

How does $v\Phi^1=\cdots=v\Phi^k=0$ imply $v\in\ker d\Phi_p$?

I'm confused about an immediate corollary in John Lee's Smooth Manifolds. Proposition 5.38 says Suppose $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold. If $\Phi\colon U\to N$ ...
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1answer
43 views

How to calculate scalar product of two gradients in indicial notation?

Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator? . $(\vec\nabla{\gamma})\cdot(\vec\nabla{\gamma})$ = ?
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0answers
35 views

differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
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0answers
15 views

Mirror of Sine-Gordon Equation and Chebychev Net

A Chebychev net and constant negative Gauss curvature $-\frac{1}{a^2}$ surfaces are described by the Sine-Gordon Equation: $2 \psi'' = \frac{\sin (2 \psi)}{ a^2}$, $\psi' = \frac{\sin(\phi)}{c}$, ...
4
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3answers
89 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
4
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1answer
510 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
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1answer
43 views

How can we find the arc length of the curve? [closed]

How can I find the length of the curve $$\left(\frac{t^3}{3} - t\right)\mathbf{i}+ t^2 \mathbf{j}, \quad 0≤t≤1?$$
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1answer
41 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
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2answers
56 views

Surfaces of Revolution with Constant Gaussian Curvature

Surface $S$ is parametrized by $$X(u,v) = (\varphi(v) \cos{(u)}, \varphi(v) \sin{(u)}, \psi{(v}))$$ with everywhere-constant Gaussian curvature $K$. Let $v$ be the arc length of the generating curve ...
7
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If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
3
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1answer
29 views

Geometric interpretation of $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$

Let $\alpha(s)$ be a smooth curve parameterised by arc-length and for fixed $r > 0$ define $\beta_r(s) = \alpha(s) + r\mathbf{n}(s)$, where $\mathbf{n}(s)$ is the unit normal vector to $\alpha$ at ...
2
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72 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
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2answers
109 views

How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China ...
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1answer
31 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...
2
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1answer
45 views

Unit circle can't be covered by one chart

I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
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0answers
36 views

Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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1answer
38 views

Why does $S^n$ satisfy the local $n$-slice condition? From Lee's Smooth Manifolds.

Example 5.9 on page 103 of Lee's Smooth Manifolds says the following: The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function $$ ...
2
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0answers
34 views

A representation of a 1 form

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^{3}$. Consider the 1-form on $\mathbb{R}^{3}$ given by $\phi = dx+ydz$. Do there exist smooth functions $u$ and $v$ such that $\phi=u\ dv$? Why? ...
2
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1answer
33 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
3
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0answers
35 views

Voisin's proof of Ehresmann's theorem

On p.221 of Voisin's book on Hodge theory, there are two claims: a) Let $B$ be a contractible smooth manifold. There exists a vector field $\chi$ on $B$ whose flow $\Phi_t$ is global and, given any ...
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0answers
38 views

Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a ...
1
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3answers
108 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
0
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1answer
29 views

How to find the surface element for the cylinder $x^2 + y^2 = r^2$?

So if given a surface (cylindrical) which has radius r and equation $x^2 + y^2 = r^2$, I want to work out the line element for it. How do I get it? I know the final answer has to be $dS^2 = r^2dϕ^2 ...
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0answers
15 views

Mean curvature of even order

I read Antonio Ros, Compact Hypersurfaces with Constant Higher Order Mean Curvatures,1987. I don't understand following sentence from the second page 6th line. From the Gauss equation, we have ...
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23 views

A problem fro do carmos differential geometry book

A half-line $[0, \infty)$ is perpendicular to a line $E$ and rotates about $E$ from a given initial position while its origin $0$ moves along $E$. The movement is such that when $[0, \infty)$ has ...
2
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1answer
34 views

Why is the inverse of the standard charts on $\mathbb{R}P^n$ continuous?

When showing that $\mathbb{R}P^n$ is a topological manifold, the atlas is given by charts $\varphi_i\colon U_i\to\mathbb{R}^n$, where the $U_i$ are the classes $[x^1,\dots,x^{n+1}]$ with $x_i\neq 0$, ...
2
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1answer
84 views

Codifferential of a $p$-vector in components

I'm learning differential geometry from a textbook, and I got stuck on a problem. I'm supposed to calculate this for a $p$-vector $F$ in $n$ dimensions: $(\mathrm{div}_\omega F)^{i...j} = ...
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1answer
102 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
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0answers
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curvature of a plane curve

I'm trying to prove the formula to calculate the curvature of a plane curve. But I end up with the wrong sign and can't figure out why: I want to proof $\kappa(t) = \frac{\dot c(t) \cdot \ddot ...
0
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1answer
35 views

Local isometric embedding

Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in ...
3
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1answer
125 views

Product neighborhood theorem with boundary

The Product Neighborhood Theorem states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization of the normal bundle (wrt. some smooth ...
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1answer
29 views

$\frac {\partial x}{\partial u} \wedge \frac {\partial x}{\partial v} \ne 0$

Let $x(u, v)$ be a parametrization of a surface $S \subset \mathbb{R}^3$. Verify that $dx_q : \mathbb{R}^2 \to \mathbb{R}^3$ is one-to-one if and only if $\frac {\partial x}{\partial u} \wedge \frac ...
3
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1answer
54 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
0
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1answer
81 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
10
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2answers
333 views

Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
2
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1answer
78 views

Literature on Chern-Weil Theory and the Chern-Gauß-Bonnet Theorem

At my university there are plans for a graduate seminar on Chern-Weil Theory and Chern's generalisation of the Gauß-Bonnet Theorem. Unfortunately I am having a though time in finding adequate and ...
10
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2answers
158 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of mine has only a bachelors degree in pure math and got hired by ...
10
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2answers
236 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
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0answers
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Gauge covariant derivative on principal bundle over $\mathbb R^d$

I try to understand the physical definition of covariant derivative in gauge theories in terms of the exterior covariant derivative of vector-valued forms defined as the horizontal projection wrt a ...