For questions about smooth manifolds.

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3answers
42 views

The Euclidean Metric on $\mathbf R^3$ Induces an Index-Lowering Isomorphism $b:\mathfrak X(\mathbf R^3)\to \Omega^1(\mathbf R^3)$.

In Lee's Introduction to Smooth Manifolds, Second Edition, the line just before Equation 14.25 reads The Euclidean metric on $\mathbf R^3$ induces an index-lowering isomorphism $b:\mathfrak ...
3
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1answer
40 views

How small is Diff(M) compared to Homeo(M)?

Let $M$ be a smooth manifold. Is it always true that the group of diffeomorphisms is strictly contained in the group of homeomorphisms? (I know this is true for $\mathbb{R}^n$, but that is only a ...
5
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1answer
52 views

Lie bracket and flows on manifold

Suppose that $X$ and $Y$ are smooth vector fields with flows $\phi^X$ and $\phi^Y$ starting at some $p \in M$ ($M$ is a smooth manifold). Suppose we flow with $X$ for some time $\sqrt{t}$ and then ...
0
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1answer
39 views

$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...
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0answers
26 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
4
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0answers
36 views

For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic

PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1 The following is my idea: First, choosing an arbitrary open coordinate ...
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0answers
36 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
0
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1answer
22 views

Embedding of a smooth manifold

Let $M$ be a smooth, n-dimensional manifold. Prove that for every $k \leq n$ there exists an embedding $ \mathbb{R}^k \to M$. I'm having trouble visualising this. How can $\mathbb{R}^2$ be embedded ...
0
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1answer
35 views

Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
5
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2answers
49 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
4
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0answers
52 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
0
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1answer
26 views

$SL(n,\mathbb R)$ diffeomorphic to $SO(n) \times \mathbb R^{n(n+1)/2-1}$?

Question : How to show $SL(n,\mathbb R)$ diffeomorphic to $ SO(n) \times \mathbb R^{n(n+1)/2-1}$? Also, how to show $SL(n,\mathbb C)$ diffeomorphic to $ SU(n) \times \mathbb R^{n^2-1}$? I have ...
4
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1answer
38 views

Parametrizations and coordinates in differential geometry - what's the difference?

From what I've read one can introduce the notion of a tangent vector to a point on a manifold in terms of an equivalence class of curves passing through that point (the equivalence relation being that ...
0
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1answer
33 views

Manifold,affine connection,vector field

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the ...
0
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1answer
26 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
3
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1answer
26 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
1
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1answer
38 views

Show that $M$ is a differentiable submanifold

Problem. Let $f_i:\Bbb{R}^4\to \Bbb{R}, \,\, i=1,2,3,$ be defined by $$f_1(x_1,x_2,x_3,x_4) = x_1x_3-x_2^2\\f_2(x_1,x_2,x_3,x_4)=x_2x_4-x_3^2\\f_3(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3.$$ Then $M=\{x\in ...
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0answers
22 views

Local parametrizations and coordinate charts on manifolds

I have recently had discussions on related questions about coordinate charts on here which has started to clear up some issues in my understanding of manifolds. Apologies in advance for the ...
2
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0answers
35 views

Jeffrey Lee 2.11 Show there is a nice map $s:TTM \to TTM$ satisfying several properties

I'm not sure this problem makes any sense on several levels, but here is the question verbatim: Find natural coordinates for the double tangent bundle $TTM$. Show that there is a nice map $s:TTM ...
3
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1answer
20 views

Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
4
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1answer
46 views

To Reconcile Two Different Descriptions of the Dual Bundle

$\newcommand{\mc}{\mathcal}$ Let $\pi:E\to M$ be a smooth vector bundle with typical fibre a $k$-dimensional vector space $\mc V$. There are (at least) two ways to construct the dual bundle of $E$. ...
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0answers
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The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
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0answers
48 views

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ ...
1
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1answer
31 views

“measure zero” and “measurable function” on Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold (which doesn't have to be orientable). As far as I know, the metric $g$ induces a "canonical" measure $\mu$ and so one can talk about sets $U\subset M$ of measure ...
3
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1answer
48 views

Applications of Principal Bundle Construction: Vague Question

I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group. To understand them better, I am looking for some applications. Can the principal ...
0
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0answers
17 views

Better proof that vector fields on submanifolds extend globally iff submanifold is closed

I just finished my own proof of one of the problems in Lee's Smooth Manfolds, 2nd ed., but I wonder if anyone knows a better (less messy) solution. It's problem 8-15, the "Extension Lemma for Vector ...
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0answers
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Finding a path to calculate a tangent space in a matrix manifold

Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$ be in $\mathbb{R}^{m \times n}$ such that $\mathrm{rank}(A) = k$ and $A_{11}$ is a $k \times k$ invertible matrix. ...
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3answers
75 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} ...
7
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1answer
59 views

Rewriting a integral using a pullback between manifolds with different dimensions

Let $M$, $N$ be differentiable manifolds, let $f: M \to N $ be a smooth map. Let $\omega \in \Omega^{dim(N)}(N)$, a dim(N)-form on $N$. Consider the integral: $$\int_N \omega$$ We know that in the ...
0
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2answers
25 views

Why is this matrix function smooth?

Let $A$ be a real, invertible, $k \times k$ matrix, let $B$ be a real $k \times (n - k)$ matrix, and let $C$ be a real $(m-k) \times k$ matrix. How is the function $$ F:(A,B,C) \mapsto CA^{-1}B $$ ...
5
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1answer
120 views

What is the origin of the terms 'jet' and 'prolongation' in differential geometry?

I am just curious what is the reason for the terms 'jet' and 'prolongation' in differential geometry? Is there some mental imagery that these names are supposed to evoke? Or are they so-named because ...
0
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0answers
13 views

Computing the derivative of a specific matrix function

Let $A$ be a real, symmetric, invertible $k \times k$ matrix and let $B$ be a real $k \times (n-k)$ matrix. How can I compute the derivative of the function $$ F:[A, B] \mapsto B^T A^{-1} B $$ I'm ...
2
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1answer
40 views

If $\phi: M_1 \to M_2$ a diffeomorphism between diff. manifolds, prove that if $M_2$ is oriented then so is $M_1$

Let $\phi: M_1 \to M_2$ a local diffeomorphism between two differentiable manifolds $M_1,M_2$. I want to prove that if $M_2$ is orientable so is $M_1$. Attempt: In order a manifold to be orientable ...
2
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1answer
15 views

Critical points of Torus height function

I'm reading Tu's intro to manifolds. He defines a critical point of a smooth map $F:N\to M$ to be a point where the differential $F_{*,p}:T_pN\to T_{F(p)}M$ fails to be surjective. He then gives ...
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0answers
31 views

suspension foliations on thickened surfaces

I've seen this statement without proof in a peer reviewed journal and I'm looking for a proof: "If $L$ is an oriented surface with boundary($\neq D^2$), and $C$ is a designated boundary component, ...
2
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1answer
28 views

Tangent and normal spaces of submanifold of fixed-rank matrices

Let $m \geq 2$. The subset $X$ of $m \times 2$ matrices with rank $1$ is a (smooth) submanifold of $\mathbb{R}^{m\times 2}$. Let $A$ be in $X$. I know from a more general statement that the tangent ...
4
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2answers
94 views

Lie Groups/Lie Algebra - Applications?

I studied Lie Groups and Lie Algebras as part of my Masters back in the 1970s. Although very elegant and beautiful, it seemed to its own little world, I never saw the connection with other branches of ...
4
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1answer
84 views

group of diffeomorphisms of interval is perfect

Every element in $\mathrm{Diff}([0,1])$, group of diffeomorphisms of interval fixing the endpoints, can be written as a product of commutators since this group is perfect (I don't know the proof ...
2
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2answers
120 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
2
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2answers
82 views

A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
2
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1answer
38 views

Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
5
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1answer
47 views

Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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1answer
39 views

Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
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0answers
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Grassmann and Stiefel manifolds

I want to show these two objects live up to their name in the sense that they actually are manifolds. The Grassmann manifold I understand to be a generalization of projective space (everything is done ...
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1answer
47 views

Making a set into a manifold

Let $n \in \mathbb{N}$, $M$ be a set and let $\mathcal{A} = \{(\varphi_a, U_a)\}_{a \in \mathcal{A}}$ be a system of tuples so that: $U_{a} \subseteq \mathbb{R}^n$ is open for all $a$; $\varphi_a: ...
0
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1answer
23 views

Confusion on when components of a vector relative to a basis are not components of a tensor

I have been studying affine connections, parallel transport and the covariant derivative. The text I am reading defines an affine connection $\nabla$ as a map $\nabla ...
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1answer
66 views

Ricci tensor and average of a tensor

Let $(M^n,g)$ be an oriented Riemannian $n$- manifold and $g$ is a Riemannian metric on $M$ , $\mathrm{d}\sigma$ is Riemannian volume form on $S^{n-1}$ and $\text{Vol}(S^{n-1})$ is volume of ...
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0answers
13 views

Regular level set theorem

Let $F:N \rightarrow M$ be a $C^\infty$ map between two smooth manifolds both of finite dimensions and a chart $(V,\psi) = (V, x^1,...,x^m) $ centered in a point $p \in M$. Why $F^{-1}(V) $ contains ...
0
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0answers
10 views

Criterion for Smoothness of a Function Mapping Into an Exterior Power of a Vector Space.

All vector spaces are assumed to be real and finite dimensional. I am trying to show that: Let $V$ be a vector space and $f:\bigoplus^k V\to \bigwedge^k V$ be defined as $$f(v_1, \ldots, v_k)= ...
1
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0answers
21 views

Two successive isometric immersions: relation between mean curvature vectors?

Let $M_0$ be a Riemannian manifold, $M_1$ a geodesic sphere of $M_0$ and $M_2$ an isometrically immersed submanifold of $M_1$, ie: $$ M_2 \subset M_1 \subset M_0$$ Take $X \in M_2$, and: $T_2$ the ...