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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779 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
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1answer
103 views

How to find the normal plane from a tangent plane?

$$f(x,y,z)=\frac{x^2}{4} +\frac{y^2}{9} +\frac{z^2}{25}=3 $$ I found the tangent plane from this surface at $P(2,3,5)$ by using the gradient vector, $\nabla F=\langle f_x, f_y, f_z\rangle$. I was ...
6
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1answer
199 views

Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
2
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0answers
109 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} ...
3
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1answer
151 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
3
votes
1answer
122 views

Linear connection on a manifold: Math vs. Physics

I have learned some Riemannian Geometry in a strongly mathematical framework, precisely from the book "J.M.Lee - Riemannian Manifolds: An introduction to Curvature". Now I'm trying to learn ...
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1answer
1k views

Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...
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0answers
72 views

Suggestion for a good book that explain Cartan's Moving Frame and Riemannian Geometry

I'm studying Riemannian Geometry, and I'm having a lot of trouble with the book Riemannian Geometry and Differential Dorms both from do Carmo.And I would like a book with examples, calculations, if ...
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0answers
64 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
3
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2answers
199 views

Show that the curves are circle.

For example, this question is also similar to the previous question I have asked, which is the following link; How to show the curves are conics. Question: Solve the equation ...
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0answers
60 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
3
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1answer
74 views

Manifold of fixed points

Let $M$ be a smooth manifold and let $G$ be a Lie group smoothly acting on $M$. Then, under suitable assumptions (if $G$ acts freely and properly on $M$) we have a new smooth manifold $M/G$ ...
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1answer
78 views

area form of the Poincare half plane

For the upper half plane $\{(u,v)|v>0\}$, its area form is $du\wedge dv/v^2$. How to compute the area between the u axis and the curve $\alpha(t)=(r\cos t, r\sin t)$, $0< t < \pi$? Is this ...
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0answers
66 views

What do we mean by $CP^2$ with reverse orientation?

$CP^2$ and $\bar{CP^2}$ are not diffeomorphic since they have non-isomorphic intersection forms. so why do we call the latter $CP^2$ with reverse orientation? it seems like we are not just reversing a ...
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1answer
67 views

How to find a dual frame at the sphere

Let $U=\{ (\theta,\phi,r):\theta \in \mathbb{R}, \phi \in ]0,\pi[,r\gt 0\}$, how I can find the moving frame. I thougt: Consider the parametization for $U$ $$(\theta,\phi,r)\mapsto ...
3
votes
1answer
86 views

Definition of divergence of a tensor

How do you formally define the divergence of an arbitrary $(p,q)$ tensor? And what does it geometrically signify?
4
votes
2answers
144 views

Determine the fibre bundle from the cocycle

I hope someone can give me some clarifications about fiber bundle. My notes say that the cocycle determines the fiber bundle up to isomorphism without any explanation at all. In other words, I can ...
3
votes
1answer
244 views

Uniform convergence in $\mathbb{R}^2$

$a$, $b$ are $2$ points in $\mathbb{R}^{2},\rho_{n}(t)\,:\,[0,1]\to\mathbb{R}^{2}$ is a sequence of continuously differentiable constant speed curves with $\|\rho_n'(t)\|=L_n$ for all $t$ from $0$ to ...
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1answer
69 views

Are critical points fixed?

Let $M$ be a smooth manifold (compact, connected, without boundary and oriented if you wish) with a smooth action of $S^1$. Let $f:M\rightarrow\mathbb{R}$ be an invariant function $f$. I know how to ...
3
votes
1answer
227 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
2
votes
1answer
73 views

Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
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1answer
150 views

Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
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1answer
63 views

Limit of Riemannian metrics on the disk.

I'm working through Burago, Burago and Ivanov's book A Course in Metric Geometry and I'm trying to solve the following excercise: If we denote by $D^2$ the standard unit ball in $\mathbb{R}^2$, then ...
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2answers
55 views

Unitary trivialization over Riemann surfaces with boundary

I am puzzled with the proof of Proposition 2.66. in the book "Introduction to Symplectic Topology" by Salamon, McDuff. The Proposition states, that every Hermitian vector bundle $E \rightarrow ...
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1answer
79 views

is normal bundle of a manifold trivial?

If you embed a manifold $M$ in Euclidean space, is the normal bundle always trivial? Or give an example with non-trivial normal bundle.
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votes
1answer
101 views

If a closed, smooth $m-1$ form, $\omega$ is nonzero at a point, there are local coordinates $x^i$ with $\omega = dx^2 \wedge\cdots \wedge dx^m.$

This is a problem on an old qualifying exam. Let $\omega$ be a smooth, closed $m-1$ form on a smooth $m$-dimensional manifold $M$. If $\omega \neq 0$ at a point $p\in M$ then there is a coordinate ...
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1answer
330 views

The Poincare Lemma for Compactly Supported Cohomology

I´m reading the proof of The Poincare Lemma for Compactly Supported Cohomology there is a part in the proof that said in the text book Bott and Tu: $d \pi_{\ast} = \pi_{\ast} d$ in other words, ...
4
votes
1answer
305 views

surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism

Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism? I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to ...
2
votes
0answers
57 views

Group generated by several vector fields

I have two (or more) smooth and integrable vector fields $v,w$ on a smooth manifold $M$. Each generates a flow map $\Phi_v$,$\Phi_w$ that forms a single parameter Lie group of diffeomorphisms. Let's ...
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2answers
147 views

Why in differential geometry tensors are usually defined as multilinear maps?

In multilinear algebra books tensors are usually defined through the universal property. Given a family of $k$ vector spaces $V_1,\dots,V_k$ over the same field $F$ we want to construct a space $S$ ...
3
votes
1answer
244 views

For a closed plane curve, showing some inequalities.

I have a problem following : Let $\gamma:[0,T]→\mathbb{R}^2$ be a closed plane curve, i.e., a regular parametrized curve such that $ \gamma$ and all its derivatives agree at 0 and $T$. For ...
2
votes
2answers
108 views

Manifold non-orientable iff. frame bundle is connected

Let $M$ be a connected smooth manifold and $L(M):=\bigcup_{x\in M}L_xM$ its frame bundle where $L_xM:=\{(v_1,\dots,v_n):\{v_1,\dots,v_n\}\text{ is a basis of }T_xM\}$. $M$ is non-orientable iff. ...
4
votes
1answer
100 views

Theorem by Whitney

For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set by a theorem due to Whitney. Could someone please point me to ...
3
votes
0answers
133 views

The relation between Dirac and Hodge-de Rham operator

On a spin manifold $M$, we can define Dirac operator \begin{equation} D: \Gamma(M,S) \to \Gamma(M,S) \end{equation} and in particular $D^-: \Gamma(M,S_-)\to \Gamma(M,S_+)$. Let us consider twist the ...
4
votes
2answers
116 views

Length of a Coastline

When B. Mandelbrot's typical example of measuring the length of a coastline is referenced, they mention how at every scale the length increases. In pure mathematics, I can imagine this quite well-- ...
2
votes
2answers
174 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
3
votes
1answer
166 views

Isometry in $\mathbb{R}^2$

Will there be an isometry in $\mathbb{R}^2$ taking the curve $\alpha(t)=(\cos(t)+1, \sin(t)+2)$, where $t\in [0,\pi]$, to the curve $\beta (t)=(t,\sin(t))$, where $t\in[0,c]$ and $c$ is a ...
1
vote
1answer
71 views

Hessian quadratic form is well defined

Could someone show why for the Hessian to be well defined ($d_{p}^{2}f(v,w) = L_{v}L_{w}f$) we need $p$ to be a critical point.
0
votes
1answer
56 views

Extending an embedding $:S^1\rightarrow \mathbb R^{n}$

Assume we have an embedding $f:S^1\rightarrow \mathbb R^n$. I want to extend $f$ to an embedding $\tilde{f}:B\rightarrow \mathbb R^n$, where $B$ is the closed unit ball of $\mathbb R^2$. In fact, I ...
4
votes
1answer
788 views

Volume form on a sphere.

Let $S^n(r)$ be the sphere of radius $r$ , $x_1^2 + ... + x_n^2 = r^2$ and let $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1} $$ Write $S^n$ for the unit ...
0
votes
2answers
63 views

Connection on complex vector bundle

Let $M$ be a $m$-dimensional Riemannian manifold. I will follow the notation of the book "From calculus to cohomology - Madsen and Tornehave" If $\xi $ is a $k$-dimensional complex vector bundle, ...
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0answers
33 views

Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
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2answers
82 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
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1answer
96 views

A (not so?) simple question about differential forms

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$. Can somebody help me, please ? Thanks :)
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0answers
99 views

Properties of a smooth bijection

What are the basic facts about a map $F: M \to N$ between manifolds (without boundary, we might specify) which is a smooth bijection? The map from $[0,1) \to S^1$ parameterizing the circle is a ...
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1answer
132 views

Existence of coordinate systems for submanifolds

I decided to do the following problem as an exercise: Let $p \in M$ be a regular point of $f: M \to \mathbb{R}$. Prove the existence of a coordinate system $(x_1,x_2,...,x_n)$ near $p$ such that ...
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1answer
103 views

Show that F(t) is an immersion

I've got here an exercise that says: "Show that the map $F:\mathbb{R}\rightarrow \mathbb{R^2}$ defined by $F(t)=(\cos t, \sin t)$ is an immersion". $F$ is an immersion if ...
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0answers
54 views

Differentiation along a curve on a manifold (Re: Schutz's intro to GR)

I am trying to show (Schutz chpt. 6 prob 13) that if two vector fields $\vec{A}$ and $\vec{B}$ are parallel transported along a curve $\gamma:\mathbb{R}\to M$ with real parameter $\lambda$ ($M$ a ...
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1answer
120 views

[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
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1answer
150 views

Orientation on boundary of a smooth open set

Suppose $D$ is an open set in $\mathbb R^n$ with smooth boundary, $bD$, that is, for each $p \in bD$, there exist a smooth function $r$: from a open neighborhood $U$ of $p$ to $\mathbb R$ such that ...