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Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

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Exercise 3 in section 2.2 on Hirsch's differential topology

This exercise says: Let $$p: M \rightarrow N$$ be a $C^s$ map and $$f: N \rightarrow M$$ a $C^r$ section of $p\ $ (i. e. $p \circ f = id_{N}$). Then we need to show: If $$1 \le r < s \le \...
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Existence of Smooth paths

How can we prove that there exists a smooth path between any pair of points in a connected smooth manifold? I think we can do this locally by smooth charts, but I don't know how to glue these paths ...
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Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
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1answer
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Explanation of defination of Manifold

While reading the book on Forms and connection ,I am stuck with following defination of manifold.I am stuck at the part after defining function $f$ for submersion. Can anyone explain me this ...
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1answer
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Disproving Submersion

Q.2 in the text. By the hint I have shown that it has 2 tangent directions. Now how does it follows that there is no submersion?
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Let $f(x,y)=(x-a)(x-b)(x-c)-y^2$ and let $M=V(f)$. How do I show that if $a,b,c$ are distinct real numbers, then $M$ is smooth manifold of dimension 1 [on hold]

Let $f(x,y)=(x-a)(x-b)(x-c)-y^2$ and let $M=V(f)$. How do I show that if $a,b,c$ are distinct real numbers, then $M$ is smooth manifold of dimension 1? Additionally, if $a=b$, what prevents $M$ from ...
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1answer
54 views

Dimension of submanifold is lees or equal to the dimension of the manifold.

Let $M \subset R^n$ be a $k$-dimensional smooth manifold. I want to show that if $N \subset M$ is a smooth submanifold of $M$, then $dim(N) \le k$. I am not really sure how to start solving this ...
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1answer
23 views

Show that $M=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2=z^2, z>0\}$ is a smooth 2-d manifold.

Let $M=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2=z^2, z>0\}$. How do I show that M is a smooth 2-dimension manifold?
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0answers
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Transversality to a family of submanifolds

Let $E$ be an $n$-dimensional smooth manifold obtained as the total space of a fiber bundle with codimension-$m$ fibers, and let $X \subset E$ be a codimension-$k$ submanifold. (More generally, one ...
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1answer
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Is there Method to visualize the object $Disc \times Disc$?

For me its clear how to build up the object $S^1 \times S^1$ , who is our old friend torus, but the product of 'interior'of these objetcs doesn't look clear to me how to build up I tried 'forget' ...
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This class of complex manifolds are real manifolds with border? [on hold]

Since the half-plane has a homeomorphism to the disc, which looks natural in complex analysis, is it possible that every complex manifold which has every map to discs be a real manifold with border ?
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1answer
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Is the set $\{ (x,\sin(\frac{1}{x}) ) | x \in (0,1)\}$ a manifold with and without the origin

Is the set $A = \{ (x,\sin(\frac{1}{x}) ) | x \in (0,1)\}$ a manifold with and without the origin. So without the origin $(0,0)$ I'm pretty sure that it is a manifold, since in each point the set is ...
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1answer
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Prove that this map is a submersion (check my work please)

There are some calculations but I would be very grateful if somebody could check whether my arguments are mathematically sound. I am new to differential geometry. You can also suggest easier solutions....
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Computing the $2^\text{nd}$ fundamental form of $\mathbb{R}_+\times M^n\to \mathbb{R}^{n+p+1}$

Let $f: M^n\to \mathbb{S}^{n+p}$ be an isometric immersion. The cone over $f$ is defined to be the immersion \begin{align*} F:\mathbb{R}_+\times M &\to \mathbb{R}^{n+p+1}\\ (t,x)&\mapsto tf(...
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1answer
133 views

Smaller neighborhood around the identity of Lie Group

This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition): Suppose G is a Lie group and U is any neighborhood of the identity. Show that there exists a neighborhood V of ...
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Show that $\int_{\mathbb{S}^{N-1}}{\Delta_{\mathbb{S}^{N-1}}u(\theta)d\theta}=0$

Show that $$\int_{\mathbb{S}^{N-1}}{\Delta_{\mathbb{S}^{N-1}}u(\theta)d\theta}=0$$ Where $\Delta_{\mathbb{S}^{N-1}}$ is the Laplace operator on the sphere $\mathbb{S}^{N-1}$. My approach: Let $\...
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1answer
35 views

Prove that the following map is a well defined symmetric bilinear map

Let $M$ be an m dimensional manifold, and $f: M \rightarrow \mathbb{R}$ be a real value smooth function and $p$ be a critical point of $f$. Let $x_p, y_p$ be vectors in $T_pM$, define a smooth ...
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1answer
71 views

Every connected orientable smooth manifold has exactly two orientations, Lee Proposition 15.9

The proof of Proposition 15.9 from John Lee's book "Introduction to Smooth Manifolds" is left as an exercise. Here is the statement: Let $M$ be a connected, orientable, smooth manifold with or ...
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1answer
36 views

A compact set in $\Bbb R^n$ with smooth boundary is a manifold?

Can a compact set $\Omega \subset \Bbb R^n$ with smooth boundary be considered as a smooth manifold with boundary?(Smooth boundary probably means $\partial\Omega$ is a $n-1$-smooth manifold?) I think ...
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53 views

Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 2

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
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Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 1

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
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37 views

Analysis prerequisites for Lee's Introduction to Smooth Manifolds?

Apart from Topology and Linear Algebra, do Spivak's Calculus On Manifolds and Zill and Cullen's DIFFERENTIAL EQUATIONS with Boundary-Value Problems (7th Edition) provide enough background for the ...
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1answer
20 views

Isometric Immersions: equivalent condition for $R^\perp=0$

In Dajczer's Submanifolds and Isometric Immersions, a couple paragraphs before presenting the Fundamental Theorem for Submanifolds, the author presents Ricci's equation: $$(\widetilde{R}(X,Y)\xi)^\...
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How is $dx^2+dy^2$ the Euclidean metric on $\mathbb R^2$

A Riemannian metric is a smooth symmetric covariant $2$-tensor field. If I put in two vectors, say $(1,2)$ and $(2,1)$, I don't get $\|(1,2)-(2,1)\|=\sqrt{2}$: $$(dx^2+dy^2)((1,2), (2,1))=2+2=4.$...
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Are properties of maps between manifolds seen in $\mathbb{C}^n$ or in $\mathbb{R}^n$ equivalent?

Let me elaborate: By properties, I mean qualities such as "is a submersion", "is an immersion", "is an embedding", etc. Let's say I have a function $f:S^1 \rightarrow S^3$ between the spheres viewed ...
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1answer
23 views

Question about manifolds with boundary

Prove that if $f:X\to Y$ is a diffeomorphism of manifolds with boundary, then $f$ maps $\partial X$ to $\partial Y$ diffeomorphically. Answer: Let $U\subset H^k$ be an open subset and let $\phi:U\...
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1answer
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Suppose that $S_1$ and $S_2$ are smooth surfaces in $\mathbb{R}^n$

(a) Let $n = 3$ and suppose that they intersect at a point $p$ and do not have the same tangent plane at that point. Show that $p$ is not an isolated point of $S_1 \cap S_2$. (b) Let $n = 4$ and ...
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Real advantage in considering germs of smooth functions

I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, ...
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Why should one think of orientation as a homology class?

Let $\pi:E\rightarrow B$ be a smooth vector bundle. I call this vector bundle to be an oriented vector bundle, if I can choose an orientation on $\pi^{-1}(b)$ for each $b\in B$ and a trivialization ...
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1answer
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Question on the proof of Sard's theorem in John Lee's book.

I tried to understand the proof of Sard's theorem suggested in John M. Lee’s “Introduction to Smooth Manifolds”. It said that $C_k$ is defined as $$ C_k := \{x \in U \vert \forall 1 \le i \le k\, \...
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1answer
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Does the set $\{ (x,y,z) | x^2-y^2 = 0, 0 \le y < z \}$ determine a manifold?

Does the set $\{ (x,y,z) | x^2-y^2 = 0, 0 \le y < z \}$ determine a manifold? From what I understand, the set looks like a triangle above the plane $xy$. However I can't present it in an ...
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1answer
29 views

The diagonals $\Delta=\{(v,v)\mid v\in V\}$ is transversal to $W=\{v,Av\mid v\in V\}$ iff $+1$ is not an eigenvalue of A

Learning Differential topology, Sorry for asking anything trivial. I am stuck in this question: Let $V$ be a vector space and let $\Delta$ be the diagonal of $V \times V$ . For a linear map $A : V ...
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1answer
64 views

Lie bracket of canonical vectors on tangent space to a point on a manifold is zero.

Let M be a manifold and $T_p(M)$ be the tangent space at $p$, and $\phi$ a local chart around $p$. Let $$\left.\frac{\partial}{\partial\phi^1}\right|_{_p},\ \cdots\ ,\left.\frac{\partial}{\partial\...
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$M$ compact, $0$ is a regular value of $f:M\to\mathbb R,$ show that $f^{-1}(0)$ is diffeomorphic to $f^{-1}(\varepsilon)$ for small $\varepsilon.$

The exact statement of the problem is: If $M$ is compact and $0$ is a regular value of $f:M\to\mathbb R,$ then there is a neighborhood $U$ of $0\in\mathbb R$ such that $f^{-1}(U)$ is diffeomorphic ...
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1answer
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The definition of the tangent vector of a manifold

In many textbooks of differential manifold, authors usually define the tangent vector on a manifold as below: Definition: Suppose $m$ is a $n$-dimensional smooth manifold, $x\in M$, the tangent ...
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0answers
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The set of differentiable functions $f:M \to \mathbb{R}$ whose domains include a given point $m\in M$ doesn't form an algebra?!

I am reading the book "Differentiable Manifolds" by Brickell and Clark. on page 54, it is written that:If we denote the set of differentiable functions $f:M \to \mathbb{R}$ on a manifold $M$ whose ...
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1answer
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Find the derivatives of a smooth function between 2 manifolds

Let $V$ be an m-dimensional vector subspace of $\mathbb{R}^{k}$, let $p \in \mathbb{R}^k$, and let $M=p+V$. Similarly, let $W$ be an n-demensional vector subspace of $\mathbb{R}^{l}$, let $q \in \...
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3answers
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The differential $df:T(M)\rightarrow\mathbb{R}$ and the differential map $df:T(M)\rightarrow T(\mathbb{R^1})$ differ by a canonical isomorphism.

This is a problem from Semi-Riemannian Geometry by Barrett O'Neill. For a smooth manifold $M$, we denote by $\mathcal{F}(M)$ the set of all smooth functions $f:M\rightarrow \mathbb{R}$, and by $T(M)$ ...
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Does the differential of a flow fix the kernel of Hessian at critical point?

While learning about degeneracy of critical points in the context of Morse theory, I formulated the following question. Possibly it is a simple consequence of the so-called "first variation" equation ...
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1answer
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Is every local diffeomorphism from $U \subseteq \mathbb{R}^k$ to a $V \subseteq M$ equivalent to a chart?

If we have a local diffeomorphism $F: U \rightarrow V$, with $U \subseteq \mathbb{R}^k$ and $V \subseteq M$, and $\mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can ...
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1answer
40 views

Why is it important to ask for injectivity and homeomorphisms when defining immersions and embeddings? What's the motivation behind it?

A smooth map $F: M \to N$ between two manifolds is said to be an immersion if the pushforward $F_{*} : T_{p} M \rightarrow T_{F(p)} N$ is injective for all $p \in M$. If $M$ and $F(M) \subset N$ are ...
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0answers
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Probability in the space of smooth vector fields

Let $(M, g)$ be a riemannian manifold. Is there a canonical/usual probability measure in the space $\mathfrak{X}(M)$ of smooth vector fields on $M$?
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2answers
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What does it mean for a smooth atlas to not be properly contained in any larger smooth atlas?

In Lee's Introduction to Smooth Manifolds, he defines a smooth atlas as maximal if it is not properly contained in any larger smooth atlas. What does this mean? Any atlas covers the manifold so it can'...
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Bounding the Euclidean distance between a given point and the “boundary” of a manifold

Consider a $d$-dimensional smooth manifold $M$ in $\mathbb{R}^D$ ($d < D$). Denote $B_r(c)$ as an open Euclidean norm ball in $\mathbb{R}^D$ with center $c$ and radius $r$. For a positive real ...
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0answers
119 views

Show that $f(v)=f(0)+\langle df(0),v\rangle_{x}+\frac{1}{2}\langle d^{2}f(0)v,v\rangle_{x}+o(\|v\|^{2})$

Let $M$ $n$-dimensional Riemannian manifold, $x\in M$ and $\phi\in C^{2}(M,\mathbb{R})$. Consider the function $f(v)=\phi(\exp_{x}(v))$ defined on a neighborhood of $0_x$ in $TM_x$. Show that, $$f(...
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1answer
23 views

The unit ball $B^4$ is homogeneous and imbedded in $\mathbb C^3$ [closed]

Let $B$ be the unit closed ball in $\mathbb R^4$. Is $B$ a solvmanifold. i.e., does there exist a solvable Lie group $G$ such that $G$ acts transitively on $B$? Can $B$ be embedded as a CR-...
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1answer
64 views

Example of two homotopically equivalent manifolds such that one admits a symplectic structure and the other does not

A smooth manifold $M$ admits a symplectic structure if there is an alternating non degenerate $2$-form $\omega \in \Lambda^2(M)$ that is also closed i.e. $d\omega = 0.$ Usually we can express ...
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5answers
57 views

The sphere $S^n$ admits a partition of unity consisting of two functions

I am reading the book "Differentiable Manifolds" by Brickell and Clark. Here is one of its problems: Show that the sphere $S^n$ admits a partition of unity consisting of two functions. I'm not sure ...
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1answer
31 views

Coordinate system of orientable manifolds must be on connected domain?

I am studying spivak's Calculus on Manifolds by myself,now a sophomore college student majoring in Mathematics. In Spivak - Calculus on Manifolds, Comments and Errata,the author said "Page 118) In ...
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0answers
33 views

Spivak manifolds - definition of $dw$ for a p-form $w$ on a manifold $M$

Spivak says the definition of $dw$ for a k-form $w$ does not make sense on a manifold because $D_j(w_{i_1, \dots , i_p})$ has no meaning. Does it have no meaning because the function w_{i_1, \dots , ...