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

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574 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
6
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2answers
51 views

Do two embeddings of a Euclidean space into a higher dimensional one only differ by a diffeomorphism?

Let $d\le n$ and $$f,g\colon\mathbb{R}^d\hookrightarrow\mathbb{R}^n$$ be two smooth embeddings. Is there a diffeomorphism $$\phi\colon\mathbb{R}^n\rightarrow \mathbb{R}^n,$$ such that $$f=\phi\circ ...
6
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3answers
77 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} ...
6
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1answer
51 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: ...
6
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1answer
179 views

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
6
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1answer
92 views

If it looks like a solid torus, walks like a solid torus, and quacks like a solid torus, is it a solid torus?

If an orientable 3-manifold $M$ has boundary the torus $S^1\times S^1$ and deform retracts to a solid torus $S^1\times D^2$, is it necessarily homeomorphic to a solid torus? Equivalently, if the ...
6
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1answer
160 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
6
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1answer
150 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
6
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1answer
286 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
6
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1answer
234 views

Finding an algebra of smooth functions on a manifold with a given product.

I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
6
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1answer
415 views

Are these definitions of submanifold equivalent?

Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
6
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1answer
380 views

Diffeomorphic Level sets Of Manifolds

Let $F:M^n \to \mathbb{R} $ be a smooth function admitting only regular values and $(M,g)$ a smooth connected riemannian manifold. I know that the vector field $ ...
6
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1answer
602 views

Is $[0,1]$ an *oriented* manifold with boundary? (and Stokes theorem)

The definitions I am using are a manifold with boundary is something locally homeomorphic to $(0,1] \times \mathbb{R}^n$ or $\mathbb{R}^n$. an oriented manifold is one where the transition functions ...
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1answer
87 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 ...
6
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1answer
160 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
6
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1answer
149 views

Tangent space for product of submanifolds

Suppose that $X_1$ is an $n_1$-dimensional submanifold of $\mathbb{R}^{N_1}$, and $X_2$ is an $n_2$-dimensional submanifold of $\mathbb{R}^{N_2}$, and let $X=X_1\times X_2$. Let $p_1\in X_1$ and ...
6
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1answer
188 views

Describing Functions on a Manifold

Well, I have a pretty simple (probably silly and basic) doubt about how do we describe functions on a Manifold. Well, just to make clear which definitions I'm using, for now what I know is that: a ...
6
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1answer
164 views

Why is $\mathbb{R} P^n$ called projective space?

I know that: If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real ...
6
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1answer
431 views

proof of stokes theorem

I don't understand the "idea" of the following proof, as well as some of the steps. As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...
6
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1answer
131 views

Tangential Space of a differentiable manifold is always $\mathbb R^n$?

Let $\mathcal M$ be a differential manifold with a point $p$. Let U be an open set, $p\in U$, on $\mathcal M$ and let $\phi,\psi:U\to \mathbb R^n$ be a charts on $\mathcal M$. I'm having diffculties ...
6
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1answer
227 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
6
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1answer
584 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
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1answer
130 views

What is a Lagrangian submanifold?

I see references to Lagrangian Submanifolds in the literature but don't know what they are. Is there a relation to Lagrangian Tori (which I also don't know what are). Could someone give a definition ...
6
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1answer
41 views

First exercise of Guillemin-Pollack. [closed]

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \dots, a_k, 0, \dots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$, considered as a subset of ...
6
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1answer
125 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
6
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1answer
90 views

Orientability of Stiefel manifold $V_2(\mathbb R^4)$

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups ...
6
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1answer
272 views

A problem from Spivak's Calculus on Manifolds

Notation As Spivak suggests, given $A\subset\mathbb R^n$, boundary $A$ denotes the topological boundary of $A$, i.e. $\overline A\cap\overline{A^c}$. Problem 5-3(a): Let $A\subset\mathbb R^n$ be ...
6
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1answer
348 views

Constant Rank theorem for domain with nonempty boundary

Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
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1answer
276 views

Understanding Proof About an Immersion

I am studying the following proof for which an excerpt is provided below: Update: I have written out a fully-detailed proof of an argument that seeks verify the claim that $\partial \psi$ is ...
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0answers
91 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
6
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0answers
118 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
6
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1answer
107 views

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define ...
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0answers
95 views

Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
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0answers
178 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
6
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1answer
195 views

Isotopy preserving inverse image $f_t^{-1}(V)$ of a homotopy

During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ...
6
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0answers
389 views

Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
6
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0answers
167 views

Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
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5answers
1k views

Why the surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
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3answers
670 views

On the Use of the Topology on Tangent Bundles

On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...
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2answers
524 views

Intuition for not-so-smooth manifolds

in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds. However, neither in ...
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2answers
210 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
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5answers
3k views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
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1answer
406 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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3answers
201 views

Bijection between $P^2 (\mathbf R)$ and planes in $\mathbf R^3$?

Is it true that there is a bijection between planes through origin in $\mathbf R^3$ and the projective plane $P^2 (\mathbf R)$? It is a remark in a lecture note I am reading. I tried to prove it but ...
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2answers
1k 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 ...
5
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4answers
288 views

Transition functions in manifold

Following is given as the definition of a manifold in a book which I am reading. Let $\{U_{\alpha}:\alpha\in I\}$ be an open covering of a Hausdorff topological space $X$ and $\phi_{\alpha}$ be ...
5
votes
4answers
192 views

Is $[0,1]$ a 1-manifold?

Is $[0,1]$ a 1-manifold? I would say no because at either endpoint the open sets containing it aren't homeomorphic to a 1-ball in $\mathbb R^1$.
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2answers
2k views

Manifold Explained

Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one. My ...
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2answers
689 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
5
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2answers
149 views

What is $S^3/\Gamma$?

Let G is a group and H is a subgroup of G. I know $G/H$ is the quotient space but I have no idea about what $S^3/\Gamma$ is, where $S^3$ is the sphere and $\Gamma$ is a finite subgroup of $SO(4)$. In ...