For questions about smooth manifolds.

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Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
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1answer
25 views

Orthonormal basis for a tangent plane

Given a manifold $M$ described by the graph of an arbitrary smooth function $f:U \subset \mathbb{R}^2 \to \mathbb{R}^3$, I would like to construct an orthonormal basis for its tangent plane $T_pM$ at ...
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29 views

there exists a unique plane in a point of a surface in $\mathbb{R}^3$ [on hold]

The question is how I can prove the existence in this problem: If $M\subset \mathbb{R}^3 $ is a smooth surface. Then, there exists a unique plane $\Gamma\subset \mathbb{R}^3$ that passes through ...
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1answer
51 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
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2answers
132 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
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1answer
125 views

Does de Rham theorem hold for manifolds with boundary?

I am following the J.Lee's book "Introduction to Smooth Manifolds", 2nd ed., page 480-486 to learn the de Rham theorem. It is proven on manifolds without boundary, which makes me curious about whether ...
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46 views

Zeros of vectorial field [on hold]

Given a $M$ manifold in ${\mathbb R}^n$ and $X:M\rightarrow TM$ a vectorial field such that $\pi\circ X=Id$ where $\pi:TM\rightarrow M$ (projection to $M$). One zero of $X$ is such that ...
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52 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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62 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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1answer
33 views

What is the difference between the terms smooth, analytical e continuous?

I saw the following (“roughly speaking”, like the author says) definition of a Lie group in ‘Group theory in Physics’, by Wu-Ki Tung: “Roughly speaking, a Lie group is an infinite group whose ...
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59 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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0answers
42 views

An Atlas for $\mathbb{R}/{2\pi \mathbb{Z}} $

I've been having some difficulty finding an atlas for $\mathbb{R}/{2\pi \mathbb{Z}}$. The way I have been thinking of this so far is by using the standard projection map $\pi$ on open intervals of ...
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1answer
39 views

Orientability of a product of smooth manifolds implies orientability of each factor

I've been learning a bit about orientability on smooth manifolds. I'm having torubles with this exercise: Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is ...
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1answer
37 views

How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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58 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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1answer
50 views

On the integrability of vector fields

Let $X$ and $Y$ be a vector field on $M$ and satisfies $[X,Y]=X$. If $X$ and $Y$ are pointwise linearly independent for some point $p$, then there is a sub manifold $N$ of $M$ such that $T_xN$ is ...
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Is it possible to reconstruct a triangulation from its $1$-skeleton?

Let's restrict to triangulations $T$ of compact and closed smooth manifolds $M$ with $\dim M=2,3$. Such a triangulation is a PL manifold homeomorphic to $M$ which geometric realization is a simplicial ...
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1answer
59 views

Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
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1answer
28 views

How to make an ideal generated by differential forms into a differential ideal?

Let $M=\mathbb{R}^4$ with standard coordinates $x_1,x_2,x_3,x_4$. Let $\alpha=x_2dx_1+x_3dx_3+dx_4$ and $\beta=2dx_2+x_1^2dx_3+x_1dx_4$ How to find a 1-form $\gamma$ such that the ideal generated ...
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1answer
45 views

Show that $T\mathbb S^1$ is diffeomorphic to $\mathbb S^1\times\mathbb R$

I just started learning Smooth Manifolds and got stuck on this question: Show that $T\mathbb S^1$ is diffeomorphic to $\mathbb S^1\times\mathbb R$ I can see that $T\mathbb S^1$ and $\mathbb ...
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1answer
68 views

When is $x\mapsto |x|^{s-1}x$ a diffeomorphism?

Consider the function $f:B^n\rightarrow B^n$ from the disk to itself $$f(x)=\vert x\vert^{s-1}x$$ where $s>0$ and we are considering the euclidean norm (we define the function to be $0$ in the ...
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1answer
52 views

Submanifolds - same dimension

Let $M$ be a smooth manifold and $N$ a closed embedded submanifold. Assume that they have the same dimension. In this case are they equal? EDIT: M is connected.
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Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
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1answer
39 views

Existence of injective function in a manifold with special atlas

I am trying do the following question: Let $M$ be a $n$-dimensional smooth manifold that admits an atlas with only two charts. Show that there exists an injective smooth map ...
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35 views

Is this cohomology isomorphic to De Rham Cohomology?

Let $(M,g)$ be a Riemannian manifold. Put $d^{*}= *d*$ where $*$ is the Hodge $*$ operator. So $d^{*}\circ d^{*}=0$. Then it introduce a (c0)homology. What is a relation between this ...
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1answer
54 views

The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
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1answer
43 views

Smooth Manifold, covered by 2 Charts is orientable if the Intersection is Connected

I came across this Question: Atlas on a smooth manifold that contains 2 charts in which Professor Lee commented that this Proposition is true only if the Intersection of the two Maps is connected, so ...
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2answers
60 views

Regular Surface: Regularity Condition

I am having some difficulty in understanding the meaning/motivation of the regularity condition in the definition of regular surfaces. The definition (restricted to $\mathbb{R}^2$ and ...
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0answers
56 views

de Rham cohomology group of $\mathbb{CP}^1$

Prove the de Rham cohomology group of the projective space $H^0(\mathbb{CP}^1, \mathbb{C})$ is isomorphic to $\mathbb{C}$ and $H^1(\mathbb{CP}^1, \mathbb{C})=0$.
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16 views

Smoothness of Product Functions

Suppose $f:\mathbb{R}^{n}\times\mathbb{R}^k\to\mathbb{R}$. Define $$f(x,y)\equiv f_x(y)\equiv f_y(x)$$ If $f_x$ and $f_y$ are both smooth, is $f$ smooth?
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1answer
21 views

Vector Bundle as a smooth manifold

Definition: It is said that a section $F:M\to E$ of a vector bundle $E$ is smooth if it is smooth as a map between manifolds. Possible Issue: A vector bundle is defined to be, a priori, a smooth ...
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1answer
20 views

$C^{\infty}$ vector field along an immersion has local $C^{\infty}$ extensions

Let $\psi:M\longrightarrow N$ be a $C^{\infty}$. A smooth vector field $X$ along $\psi$ (i.e. $X\in C^{\infty}(M,TN)$ $\textrm{and } \pi\circ X=\psi$) has local $C^{\infty}$ extenstions in $N$ if ...
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1answer
50 views

Graph theoretic view on manifold triangulations

To make the question (hopefully) clearer, I reformulated it: Some triangulation $T$ of a smooth manifold $M$ is a piecewise linear manifold, because smooth manifolds are topological manifolds. Such a ...
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1answer
102 views

Is a Variety a manifold?

Is it true that every smooth variety (over $\mathbb{R}$ or $\mathbb{C}$ ) is a (real or complex) manifold? I have tried to show this using the implicit function theorem but I am not getting anywhere. ...
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48 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
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1answer
52 views

Extension of smooth maps at a cusp

There is a short remark in deCarmos "Riemannian Geometry" (p. 67) and I wonder about the condition that the vertex angles must be $\neq \pi$. If $s_1$ and $s_2$ are two differentiable maps on an ...
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0answers
43 views

Prove the existence (or well-definedness) of the induced connection in tensor bundle

Given a connection $\nabla$ on a vector bundle $E$ over a smooth manifold $M$, we know there is a unique extension of $\nabla$ to all tensor bundles of $E$ that satisfies Leibniz rule and contraction. ...
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1answer
35 views

Graph of a continuous function is a smooth manifold? [duplicate]

Let $f:(a,b)\to \mathbb{R}$ be a continuous function and define $\Gamma(f) = \{(x,f(x)):x\in (a,b)\}$. The two maps $\Psi: \Gamma(f)\to (a,b)$ given by $(x,f(x))\mapsto x$ and $\Phi: (a,b)\to ...
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1answer
51 views

How to find Partition of unity in $ \mathbb{S}^n$ with only $2$ functions

How to find Partition of unity in $\mathbb{S}^n$ with only $2$ functions?
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1answer
73 views

How to understand blowing up a submanifold

I am trying to understand the idea of blowing up a submanifold of a smooth real manifold. The definition I know is replacing the submanifold by its unit tangent bundle (however, in the place I read ...
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1answer
34 views

Why is every tangent vector part of a vector field?

I am reading a book that defines tangent vectors and vector fields on a manifold $M$ as derivations: A vector field is defined as a linear function and derivation $C^{\infty}(M)$ to $C^{\infty}(M)$. ...
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1answer
28 views

is a differentiable function is necessarily a smooth one

i know that each smooth function is differentiable which follows from the fact that if partial derivatives exist at each point in a domian then f is differentiable everywhere on that domain. but does ...
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1answer
30 views

Is there a local flow that is diffeomorphic at any time?

This question is regarding p.223-224 of Loring Tu's Introduction to Manifolds (Second edition). Without proof, the author previously assumed the following. Theorem. Let $M$ be a manifold and $X$ be ...
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2answers
155 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
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1answer
35 views

How to show that $n$ -dimensional sphere $S^{n}$ is a smooth manifold

Let $S^{n}:=\{x\in\mathbb{R}^{n+1}\;:|x|=1\}$, then $S^{n}$ is a smooth manifold. (Spivak ''Calculus on manifolds'' page 111) Spivak's remark is ''note that $S^{n}=g^{-1}(0)$ and ...
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1answer
48 views

Is the following set a manifold?

Show (using the implicit function theorem) that the following subset $$M:=\{(x,y,z)\in\mathbb{R}^{3}\;|x^2+y^4+z^4=3\}\subseteq\mathbb{R}^{3}$$ Theorem: Let $A\subset \mathbb{R}^{n}$ be open let ...
3
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1answer
69 views

(Whitney) Extension Lemma for smooth maps

I am currently reading Lee's book "Introduction to Smooth Manifolds (2nd edition)". Corollary 6.27 in that book states that a smooth map $f : A \rightarrow M$ where $M$ is a smooth manifold ...
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2answers
117 views

Why are 'differential operators on manifolds' differential operators?

It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they ...
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1answer
48 views

Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds? Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could ...
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1answer
56 views

Proving that something is a manifold

I'm a beginner at differential geometry and I'm having some trouble with the following problem: Let $M \subset \mathbb{R}^n$ be a $k$-dimensional smooth manifold (smoothly embedded in ...