For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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3
votes
1answer
187 views

Trivialisation of the normal bundle of $S^1$

I'd like to show that the normal bundle of $S^1$ is trivial. That is, I want to find a homeomorphism $\varphi : NS^1 \to S^1 \times \mathbb R$ such that $\varphi \mid_{N_s S^1}$ is linear for every $s ...
1
vote
1answer
268 views

Stereographic projection of $S^1$ onto $\mathbb R$

Can you tell me if this is correct: I define $f_{(0,1)}: S^{1+} \setminus \{(0,1)\} \to \mathbb R$ as $ f((x,y)) =\frac{x}{1-y} $ where $S^{1+} = \{ (x,y) \in S^1 \mid y \geq 0 \}$. Since I removed ...
3
votes
1answer
408 views

Framed manifolds and framed knots

I've been looking at intersection forms and the Arf invariant recently and I got a comment to one of my previous questions related to this. So I looked at framed manifolds. There seems to be quite a ...
7
votes
2answers
315 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
2
votes
0answers
79 views

laplacian for functions problem with the integral on manifolds

I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...
5
votes
2answers
882 views

Spivak's proof of Inverse Function Theorem

I am having trouble with Spivak's proof of the Inverse Function Theorem in his Calculus on Manifolds: 2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n\to\mathbb{R}^n$ is ...
4
votes
2answers
473 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
3
votes
3answers
615 views

Questions about definition of Tangent Space

I have some questions about the definition of tangent space that arose after reading the book Differential Geometry of Curves and Surfaces of Manfredo do Carmo. First, what's the best way to define ...
3
votes
0answers
334 views

an injective immersion between two compact manifold of same dimension

$f:M\rightarrow N$ be a injective immersion, where $M$ and $N$ are same dimensional manifold with out boundary, we need to show $f$ is a covering map. what I tried is, $df_x:T_x(M)\rightarrow ...
1
vote
1answer
405 views

classification of 1-manifolds

i read that the circle $S^1$ is the only connected compact 1-manifold but don't we have that the interval $I=[0,1]$ is a connected compact 1-manifold and that is not homeomorphic to $S^1$? May be they ...
1
vote
1answer
165 views

manifold as simplicial complex

I want to know the topological relation between a manifold and a simplicial complex. I know that a simplicial complex cannot be a manifold since its a union of simplices which are manifolds of ...
3
votes
4answers
640 views

global section vector bundle

do non-zero global section always exist in a manifold $M$? If $M$ is compact I think they do because taking a partition of unity $\rho_{\alpha}$ subordinated to a finite covering, and defining local ...
6
votes
2answers
398 views

Question about definition of topological manifold

The following definition of topological manifold is given in Lee's Introduction to topological manifolds (2000) on page 33: A topological manifold is a second countable Hausdorff space that is ...
0
votes
1answer
143 views

Dimension of disjoint union of manifolds

While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_1$-manifold and a $d_2$-manifold is still a manifold and if yes ...
3
votes
0answers
52 views

Show that the cosets of a closed isotropy group form a manifold

Suppose $G$ is a Lie group acting on the manifold $M$ and $p \in M$ is such that $G_p$, the isotropy group of $p \in M$, is closed in $G$. I'm trying to prove that $G/G_p$ has a manifold structure. ...
5
votes
1answer
242 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
1
vote
0answers
100 views

Neighborhood Retraction of Boundary

Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$. I realize that the collar neighborhood ...
1
vote
0answers
55 views

A K-invariant submanifold of G-manifold and fundamental vector fields

Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$ $$ \underline{X}(p) := ...
2
votes
0answers
259 views

Why $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded submanifold?

How can I prove that $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded smooth($C^\infty$) submanifold of $\mathbb{R}^2$. I tried to say there is any ($C^{\infty}$) immersion from $\mathbb{R}$ into ...
1
vote
0answers
92 views

extension of the exterior derivative

Is it true that since the smooth forms are dense in the $L^2$ and $H^{1,2}$ sections of forms, we can extend the exterior derivative $d$ and its adjoint $d^*$ to this spaces?
0
votes
1answer
133 views

$L^2$ definition on a manifold

Is the space of $k$-forms on a compact Riemannian manifold $M$ with the inner product given by $$(\alpha,\beta)=\int \alpha \wedge *\beta=\int g(\alpha,\beta)dv$$ which is called «$L^2$ product in ...
1
vote
2answers
180 views

Manifold/Topology Notation

I have a basic notation related doubt as follows: Let $M\subset \mathbb{R}^N$ be a manifold. What does $C^\infty(M)$ denote in $f \in C^\infty(M)$?
2
votes
2answers
69 views

Is it true that $R^2 = E(2)/U(1)$?

(Just so we're clear: that the Lie group of planar translations $R^2$ is isomorphic to a quotient of the 2D Euclidean Lie group $E(2)$ and the circle group $U(1)$.) I am trying to prove that $R^2 = ...
3
votes
1answer
117 views

A question on Lie sub-group

Well, definition of Lie subgroup what I know is, a Lie subgroup of a Lie group $G$ is an abstract subgroup $H$ which is an immersed submanifold via the inclusion map so that the group operations on ...
3
votes
1answer
201 views

Invariance of Topological Manifold Dimension: $1$ Dimensional Case

I am currently practicing with a few problems in Lee's Introduction to Topological Manifold, and I decided the complete the following problem: Prove that a nonempty topological space cannot be ...
3
votes
1answer
122 views

Every $1$-manifold is orientable

How to prove that every $1$-manifold is orientable? Can I use Zorn's Lemma and produce a maximal orientable manifold that will have to be all M?
0
votes
2answers
143 views

Derivative as derivative around zero?

Am I right that I can write/interpret any derivative $\frac{\partial f(x)}{\partial x}$ as derivative around zero, i.e.: $$\frac{\partial f(x)}{\partial x}=\left.\frac{\partial f(h+x)}{\partial ...
3
votes
1answer
99 views

Wedge Sum of Axes Not a Manifold

Note: for this question a topological manifold is defined to be a locally Euclidean, second countable, Hausdorff space. Also, I am using the subscript $x$ and $y$ just to keep track of which copy of ...
3
votes
1answer
734 views

Geometric meaning of a nondegenerate critical point

Let $f\!:M\!\rightarrow\!\mathbb{R}$ be a smooth function on a manifold and $p\!\in\!M$. Is there any way to geometrically/visually characterize the conditions $p$ is a critical point (i.e. ...
3
votes
3answers
198 views

Class of manifolds is a set?

Is the class of all 2-countable manifolds a set? I think so: each such space is a countable union of sets of cardinality $|\mathbb{R}^n|\!=\!|\mathbb{R}|$, i.e. a manifold has cardinality continuum, ...
4
votes
1answer
109 views

Equivalent form of definition of manifolds.

I am studying topology on my own, and I am having trouble proving the following. For a Hausdorff, connected, locally euclidean paracompact space $X$, there exists a countable basis for $X$. I ...
1
vote
1answer
133 views

Manifold Topology and Geometry

I am green in manifold and I have some geometry background. I have some question about "manifoldness" of a triangular mesh. To my understanding, charts {$\phi_{\alpha},U_{\alpha}$} and transition maps ...
43
votes
1answer
1k views

What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
3
votes
2answers
220 views

Integration of a differential form

Let $\omega$ be a $2$-form on $\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}$, $$\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+ ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge ...
4
votes
1answer
134 views

Boundary Question in $\mathbb{R}^{2}$ (Manifolds)

Given a subset $A$ of $\mathbb{R}^{n}$, a point $x \in \mathbb{R}^{n}$ is said to be in the boundary of A if and only if for every open rectangle $B\subseteq\mathbb{R}^{n}$ with $x\in B$, $B$ contains ...
3
votes
1answer
548 views

Regular value: intuition about surjectivity condition

Let $f:M\rightarrow N$ be a smooth function between two smooth manifolds. A $\textit{regular point}$ is a point $p\in M$ for which the differential $df_p$ is surjective. What does the surjectivity ...
3
votes
3answers
385 views

Prerequisites for studying smooth manifold theory?

I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to ...
3
votes
2answers
351 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
1
vote
1answer
92 views

Tangent field of a strictly convex closed curve

Let $\gamma:[0,a] \to \mathbb{R^2}$ be a simple smooth closed curve with curvature $\kappa (t) \neq 0$ $\forall t \in [0,a]$. Prove for each $\vec{u} \in S^1$ there exists a unique $t_0 \in [0,a]$ ...
3
votes
2answers
304 views

Does a diffeomorphism between manifolds induce an isomorphism of Sobolev spaces?

Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less ...
3
votes
3answers
466 views

Boundary of product manifolds such as $S^2 \times \mathbb R$

Simple question but I am confused. What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$? What would be the general way to evaluate the boundary of a product manifold? Thanks for the ...
5
votes
0answers
125 views

Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M ...
8
votes
5answers
636 views

Uncountable disjoint union of $\mathbb{R}$

I'm doing 1.2 in Lee's Introduction to smooth manifolds: Prove that the disjoint union of uncountably many copies of $\mathbb{R}$ is not second countable. So first, let $I$ be the set over which we ...
1
vote
1answer
102 views

Why is the fundamental group of a prime, reducible 3-manifold $\mathbb{Z}$?

I've read in a paper that if $M$ is a prime, reducible $3$-manifold, then $\pi_{1}(M) \cong \mathbb{Z}$. Can anyone explain why this is true? Thanks in advance.
1
vote
0answers
229 views

Constructing a Tangent Space Functor

This is a follow up of this question I asked some time ago regarding the tangent space functor. I am wondering though if there is a simpler way that this situation can be characterized without ...
2
votes
1answer
267 views

Two Lie algebras associated to $GL(n,\mathbb{C})$

I have elementary questions about Lie groups and their associated Lie algebras. Let $G=GL(n,\mathbb{C})$. Then associated to this Lie group is the Lie algebra $M_n(\mathbb{C})$ with the commutator ...
1
vote
0answers
150 views

If $f:M\rightarrow N$ is $C^{\infty}$, bijective, and everywhere non-singular, then $f$ is a diffeomorphism

I am not able to solve this problem: Prove that if $f:M\rightarrow N$ is $C^{\infty}$, one-to-one, onto, and everywhere non-singular, then $f$ is a diffeomorphism. This $f$ is a diffeomorphism ...
2
votes
2answers
245 views

Differentiable structure on the real line

The usual differentiable structure on real line was obtained by taking ${F}$ to be the maximal collection containing the identity map, Let ${F_1}$ be the maximal collection containing $t\mapsto t^3$. ...
4
votes
1answer
370 views

$M/\Gamma$ is orientable iff the elements of $\Gamma$ are orientation-preserving

The probem is: Suppose $M$ is connected, orientable, smooth manifold and $\Gamma$ is a discrete group acting smoothly, freely, and properly on $M$. We say that the action is ...
2
votes
0answers
88 views

flows on a manifold and liebracket

I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows ...