For questions about smooth manifolds.

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Embedding of a finite group in a compact connected Lie group

How can one embed a finite group $G$ in a compact connected Lie group? I think if we take a faithful unitary representation of G , that will do the job.But if $G= Z/n$, then what should be the ...
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1answer
28 views

The Riemannian Distance function does not change if we use smooth paths?

The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all piecewise smooth paths between $p$ and $q$. Does it change if we take the infimum only over smooth ...
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1answer
48 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
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1answer
42 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
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Transformation of a subset of compact Jordan sets to manifolds

Let $T$(for e.g. $[0,1]^2$) be a Jordan compact sets and $\tau$ be a "smooth enough one-to-one" transformation, i.e.($\tau: [0,1]^2 \rightarrow [0,1]^2 $). Lets take a subset of Lebesgue measure ...
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1answer
60 views

In manifold theory, in what sense is the derivative a first-order approximation?

As I move on from the calculus definition of the derivative to the differential geometric definition in terms of tangent spaces, I am wondering how to recover the notion that the derivative of a ...
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2answers
64 views

Vector fields along maps: I need another sanity check

Consider the definition of a vector field along a smooth map $f: M \to N$ where $M,N$ are smooth manifolds: A vector field along $f$ is a continuous map $W \colon M \to TN$ such that $W(m) \in ...
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2answers
35 views

Finite universal covering induces injective maps on cohomology

I am trying to prove the following: Suppose $M$ is a smooth, connected manifold with finite fundamental group and $f : \widetilde{M} \rightarrow M$ is its (smooth) universal cover. Show that $f^* : ...
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82 views

Tangent space manifold

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$. I have ...
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1answer
20 views

Non-existence of local inverse for smooth map $S^3 \rightarrow \mathbb{R}^3$.

I am working on a problem which asks me to show that for any smooth map $f : S^3 \rightarrow \mathbb{R}^3$, there must exist at least one point $p \in \mathbb{R}^3$ where there fails to exist a local ...
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2answers
47 views

Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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1answer
86 views

How to see that SL(2,C) is simply connected?

I started reading about Lie groups and right now I'm trying understand why $SL(2,\mathbb{C})$ is simply connected. I have shown that $SU(2)$, being diffeomorphic to $S^3$, is simply connected. So my ...
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1answer
47 views

Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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1answer
52 views

Volume of Manifold with zero Lebesgue measure

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...
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1answer
39 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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1answer
54 views

A Map into the Tangent Bundle Which Can be Restricted to the Tangent Bundle of a Submanifold

$\newcommand{\R}{\mathbf R} \newcommand{\wh}{\widehat} \newcommand{\vp}{\varphi}$ I think the following should be true. Let $S$ be an embedded $k$-submanifold of a smooth $n$-manifold $M$ and let ...
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2answers
37 views

Whitney Embedding Theorem

This is very basic question, but from my previous question I learnt that "Whitney Embedding Theorems states that any smooth n dimensional manifold can be embedded in Euclidean space of dimension at ...
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1answer
40 views

Geodesic vector field is well-defined

Let $(M,g)$ be a Riemannian manifold. I just learnt that for a curve $x:I\to M$ to be a geodesic, the geodesic equation $$\ddot{x}^k+\dot{x}^i\dot{x}^j\Gamma^k_{ij}=0$$ is equivalent to the condition ...
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1answer
52 views

Flowout Theorem

I am reading Theorem 9.20 (Flowout Thoerem) from Lee's Introduction to Smooth Manifolds, Second edition. A part of the theorem states the following: Let $M$ be a smooth manifold and $S$ be a ...
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1answer
71 views

Is $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$ a differentiable submanifold?

Let $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$. Determine whether or not $M$ is a differentiable submanifold. I honestly couldn't get anything out of it. What is the standard approach to this ...
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1answer
55 views

$SL(n)$ is a differentiable manifold

Prove that $SL(n)=\{A\in \Bbb{R}^{n\times n}:\det(A)=1\}$ is a differentiable submanifold. The determinant function is smooth since it's a polynomial, and we have $\det^{-1}(1)=SL(n)$. So it ...
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15 views

Subset of Jordan set of positive lebesgue measure

let $T \subset \mathbb{R}^d$. Given on $T$, a Jordan set of positive Lebesgue measure, $l(T)>0$ . Let a set $M \subset T$; with $l(M)=0$. Please explain what is special about the set M. Has it got ...
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1answer
62 views

What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
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Given an embedded $C^k$ submanifold of $R^n$, is the distance function $C^k$?

I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient. Let me be a little formal for clarity: Let $M$ be a ...
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Jeffrey Lee 5.17 Equivalent conditions on commuting left-invariant vector fields on $GL(V)$

Here is the question: Let $A,B \in \frak gl$$(V)$, where $V$ is a finite-dimensional vector space over the field $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, and show that the following statements ...
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1answer
35 views

Showing a vector field is smooth.

Let $(M,g)$ be a Riemannian manifold, $N$ a smooth manifold and $$\pi:M\to N$$ a surjective smooth submersion. Then, each level set $M_q=\pi^{-1}(q)$ is a properly embedded submanifold of $M$ so we ...
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1answer
44 views

Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$

I think I managed to show this statements but I am not sure about it. Since this is common problem in differentiable manifolds I was wondering if anybody has (or may write) a solution. Let $X$ be a ...
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40 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
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Is there a natural map $\Omega^1_X\to N^{\vee}_{Y/X}$?

Let $X$ be a smooth complex manifold and $Y\subset X$ be a complex submanifold. Is there some natural map from $\Omega^1_X$ to $N^{\vee}_{Y/X}$?
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1answer
31 views

Continuity in definition of Induced Functional Structure

I have a really simple question, however I am confused. Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows: Suppose $F_x$ is a functional structure on space ...
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3answers
50 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 ...
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1answer
44 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 ...
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1answer
56 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 ...
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1answer
44 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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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 ...
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1answer
84 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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45 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 ...
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1answer
29 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 ...
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1answer
45 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 ...
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2answers
53 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 ...
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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 ...
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1answer
34 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 ...
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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 ...
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1answer
47 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 ...
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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 ...
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1answer
30 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 ...
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40 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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25 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 ...
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38 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 ...
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1answer
23 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 ...