For questions about smooth manifolds.

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1answer
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Tensor fields on a manifold

Let $M$ be an $n$-dimensional smooth manifold. It is easily shown that the modules $\Gamma(TM)$ (the real vector space of vector fields on $M$) and $\Gamma(T^\ast M)$ (the real vector space of $1$-...
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38 views
+100

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
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1answer
32 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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22 views

Invariance of Linking numbers and critical values

So, I am trying to show that for a map $f: S^{2p-1} \rightarrow S^p$ , the linking number $l(f^{-1}(y),f^{-1}(z))$ of two framed submanifolds associated with regular values $y,z$ of $f$, defined as ...
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1answer
35 views

The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle.

Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}...
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0answers
17 views

Differentiable sub manifolds and regular parametrization

Let $0<r<R$. Consider the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \mid (\sqrt{x^2+y^2}-R)^2+z^2=r^2\}.$$ How can I show that $T^2$ is a two-dimensional differentiable submanifold of $\mathbb ...
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1answer
76 views

What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
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1answer
47 views

Frobenius condition in terms of Lie brackets

Let $\alpha$ be a $1$-form and $\xi = \ker \alpha$. Frobenius theorem tells us that $\xi$ is integrable iff $\alpha\wedge{\rm d}\alpha = 0 .$ In the book "Introduction to Contact Topology" from ...
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1answer
32 views

Compact differentiable sub manifold with at least two points [closed]

Let $M$ be a differentiable submanifold of $\mathbb R^n$ which contains at least two points. How can I show that if $M$ is compact in $\mathbb R^n$ there exists no atlas for $M$ which only consists of ...
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2answers
109 views

The integral of a function on manifold and differential form

When we want to integrate a function f over a manifold M, we may meet some problems, for example, the problem showed in the picture below: Then people used differential form to integrate. But it ...
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20 views

Ring of smooth functions on a manifold and localization with respect to a multiplicative system

Take $X$ a smooth manifold and $x\in X$. It can be shown that the germ of smooth functions around $x$, $C^\infty(X)_x $ is equal to the algebraic $S^{-1}C^\infty (X)$ where $S$ is the set of smooth ...
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1answer
64 views

A question about pullback of the K-form

Let $M$ be an oriented $m$-dimensional manifold. Suppose the support of $\omega$ is in an open subset $U$ of $M$, and $\phi \colon U \to R^m$, $\psi \colon U \to R^m$ are two different charts on $M$ ...
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1answer
34 views

Top cohomology of a non-orientable smooth surface with boundary.

I would like to know what the singular relative cohomology $H^2(M,\partial M;\mathbb{Z})$ of a smooth connected surface with boundary $M$ is. In the orientable case I did the following: The zero-th ...
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0answers
37 views

Is the space of rigid Riemannian metrics convex?

Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity). Is it true that every convex combination of $g_1,g_2$ is also rigid? ...
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1answer
56 views

Understanding Milnor's proof of the fact that the preimage of a regular value is a manifold

In the book "Topology from the Differential Viewpoint" (Milnor) he proves on page 11 the following lemma: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is ...
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21 views

Is the Inverse of a Coordinate System on a Differentiable Manifold Differentiable?

In his book Calculus on Manifolds, Spivak shows that a differentiable k-dimensional manifold in $\mathbb{R}^n$ is a set $M$ such that $\forall x \in M$ there exists an open set $U$ containing $x$, an ...
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113 views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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34 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
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1answer
42 views

Geodesic curvature of a curve in the hyperbolic plane

Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$ Calculate the geodesic curvature of $\gamma$. The problem I'm ...
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32 views

coordinate functions relation with covectors

I think that this should not be a difficult question to answer but I couldn't solve it by myself, so here is the question: Let $f_{1}, ..., f_{r}$ be $C^{\infty}$ functions on an open set $U$ of a ...
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1answer
28 views

$G$-structure of a product manifold

My question concerns $G$-structures on manifolds: Let $M$ be an $n$-dimensional manifold. Since any $n$-dimensional manifold admits a riemannian metric, $M$ admits an $O_n$-structure. Similarly, $\...
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2answers
122 views

Is $n$-dim manifold which is immersed in $\mathbb{R}^n$ diffeomorphic to a ball?

Let $M$ be an $n$-dimensional smooth compact oriented manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ...
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37 views

Making a principal bundle into a covering space

Suppose $\pi : P\rightarrow M$ is a principal $G$-bundle. I want to make this into a covering map by changing the topology of $P$. By local triviality we can find for each $x\in M$ an open $U\subset ...
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2answers
74 views

Show that geodesic equation is given by $\ddot x^k +\Gamma_{ij}^k \dot x^i\dot x^j=0$

I know that $\gamma $ is a geodesic if and only if $$\nabla _{\dot \gamma}\dot\gamma =0.$$ Using this, I'm trying to re find the equation $$\ddot x^k +\Gamma_{i\ell}^k \dot x^i\dot x^\ell=0,$$ but I ...
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1answer
51 views
+50

Why $\nabla _{\dot \gamma (t)}Y_t=\dot x^i \frac{\mathrm d a^j(t)}{\mathrm d t}\partial _j+\dot x^i a^j\nabla _{\partial _i}\partial _j$

Let $M$ a smooth manifold and $\nabla $ a connexion. Let $\gamma :[a,b]\longrightarrow M$ a $\mathcal C^\infty $ curvature. I recall that if $X,Y\in \Gamma(M)$, and $f,g\in \mathcal C^\infty (M)$, ...
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0answers
34 views

geometric interpretation of eigenfunctions of a vector field

Let $M$ be a smooth manifold and $X\in\mathfrak X(M)$ be a section on the tangent bundle. What is the geometrical interpretation of the eigenfunctions of $X$. That is functions in $f\in \mathcal C^\...
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1answer
29 views

What is the smoothness of a family of diffeomorphisms $t\mapsto \psi_t \in \text{Diff}(M)$ ? And how to interpret it intuitively?

First of all, we have to give the group $\text{Diff}(M)$ of all diffeomorphisms on $M$ a smooth-manifold structure. (To see this, it may be helpful to consider a easier problem: how to give the group $...
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2answers
37 views

Union of submanifolds

Let $M$ be a smooth manifold (without boundary) and $A,B$ too submanifolds of $M$ such that $$A\cap B=\emptyset\quad\text{and}\quad\dim A=\dim B.$$ Is $A\cup B$ a submanifold of $M$? The assumption ...
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1answer
20 views

Can I conclude $s$ is a submersion from these data?

Let $M$ and $N$ be smooth manifolds ($C^\infty$). Let $s\in C^\infty(M, N)$ and $u\in C^\infty(N, M)$ be maps satisfying: $u$ is an embedding; $s\circ u=\textrm{id}_N$; $(u\circ s)^2=u\circ s$. ...
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0answers
27 views

Inner products on $C_c^\infty$ corresponding to differential forms

Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a ...
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1answer
51 views

Normal variation of embedded surfaces

Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by $$\phi(p,t)=\exp_p(...
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1answer
60 views

Is every smooth manifold the solution set of finitely many equations?

In the case of affine varieties we have finitely many (polynomial) equations defining the variety. It is a (smooth) manifold iff it satisfies the Jacobian criterion. I wonder whether this can be ...
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1answer
24 views

A basic question about this identity in Lie group setting.

$\textbf{Problem.}$ Let $G$ be Lie group. Let $F:G\times G\rightarrow G$ denote the multiplication map. Identify the space $T_{(e,e)}(G\times G)$ with $T_{e}G\oplus T_{e}G$ by $$v\in T_{(e,e)}(G\...
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1answer
46 views

Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
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1answer
115 views

Poincare lemma for compact vertical supports in Bott & Tu

I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the ...
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0answers
6 views

Using partition of unit to extend paths on vector bundles in time dependent sections.

Let $\pi: A\longrightarrow M$ be a vector bundle and $a: I\longrightarrow A$ be a path where $I=[0, 1]$. I would like to show there are time dependent section $\alpha: I\times M\longrightarrow A$ ...
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2answers
247 views

Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-...
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Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
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1answer
52 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
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31 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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0answers
20 views

Boundary of a sublevel

Let $f: B_1(0)\subset\mathbb{R}^2 \to \mathbb{R}$ be a smooth function, and say $c\in \mathbb{R}$ is not a critical value of $f$. Is it true that each connected component $\Gamma$ of $\left\{f\le c\...
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1answer
25 views

Construct vector field along a curve

Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along $c|_{[...
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2answers
48 views

Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?

Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an exact ...
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2answers
25 views

Can a Submanifold Become Tangent to a Nowhere Tangent Vector Field

$\newcommand{\R}{\mathbf R}$ Let $M=\R^2$ and $S=\{(0, t):-1<t < -1\}$ be a submanifold of $M$. Let $V$ be a vector field on $M$ which is nowhere tangent to $S$. Let $\theta$ be the flow of ...
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3answers
119 views

What are interesting corollaries of a manifold being parallellizable?

This is a heavily edited (in fact, a complete rewrite) of a question I asked badly a few days ago. I am editing as opposed to asking a new question as there are already several relevant answers. I ...
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1answer
45 views

Show that a smooth manifold modulo diffeomorphism group is a smooth manifold

Would like help in starting this exercise: Suppose $\Gamma$ is a group of diffeomorphisms of a manifold $\left( {X,C_X^\infty } \right)$. Suppose that the action of $\Gamma$ is fixed-point-...
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1answer
22 views

constructing a manifold structure for a plane in $\mathbb{R}^3$ [closed]

Any help on this problem would be greatly appreciated. thanks! Let M be the plane in $\mathbb{R}^3$ with normal vector (a,b,c)$\neq$0. Construct a manifold structure each topological space (M,$\tau$) ...