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

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20 views

Differential identity and wedge products

Apparently $dx^{i_1} \wedge ... \wedge dx^{i_k}=d(x^{i_1}dx^{i_2}\wedge ... \wedge dx^{i_k})$ which I cannot see proved anywhere in my notes. It just stated as if it is obvious which I don't believe ...
2
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2answers
58 views

Part of proof that $d^2\omega=0$

The following comes from the proof in differentiable manifolds that $d^2\omega=0$. Let $f$ belong to the set of $0$-forms. From definition I have that $\displaystyle df = \frac{\partial f}{\partial ...
3
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3answers
46 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
4
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1answer
52 views

Existence of critical points of $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$

I am trying to show that an smooth, proper map, $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$ has a critical point. My attempt was to suppose there are no critical points, then the preimage of every point is ...
6
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1answer
90 views

Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
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2answers
30 views

Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...
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1answer
34 views

Every solution of the system is attracted to the center manifold

I am trying to solve the following problem. Determine a center manifold for the rest point at the origin of the system \begin{align} \dot x &=-xy \\ \dot y&= -y+x^2-2y^2 \end{align} a) ...
4
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1answer
53 views

The system $\dot{x}=x^2$, $\dot y=-y$, has infinitely many (local) center manifolds

Consider the system, \begin{align} \dot{x}&=x^2 \\ \dot y&=-y \end{align} I am trying to show that this system has infinitely many local center manifolds. Here is what I have done so far: ...
4
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1answer
96 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
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0answers
12 views

Math Bases of Comparison and Association [closed]

My question is about the cognitive phenomenon of intuitive pattern-matching or association by similarity / dissimilarity. Imagine a situation where a person has a particular experience, which might ...
2
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1answer
36 views

Bogus proof that the Liouville Form on the cotangent bundle is nondegenerate.

Suppose we have a manifold $M$ of dimension $n$ and its cotangent bundle $T^*M$. The Liouville form $\lambda$ on $T^*M$ is defined as $\lambda_{\omega_p} = \pi^*(\omega_p)$ where $\pi$ is the standard ...
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1answer
35 views

Is stereographic projection the only way to make a bijection between plane and sphere?

At a math exhibition, I learned the concept of stereographic projection for the first time. However, I am curious about the purpose of the stereographcal projection. I've learned that an area of ...
1
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1answer
41 views

Any two $1$-dimensional manifolds in $\mathbb R^3$ can be made disjoint by translating one of them

I'm trying to solve this innocent problem. Let $X,Y\subset \mathbb{R}^3$ be two 1-dimensional manifolds. Show that there exists $v\in \mathbb{R}^3$ such that $X$ and $Y+v$ are disjoint. I know ...
3
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0answers
59 views

Manifolds of Non-Standard Dimension

Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many ...
2
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0answers
32 views

Compute $[\Lambda,\ \bar{\Lambda}]$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
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0answers
41 views

Using an induction on $q$.

I have a problem (It's a Proposition in Baouendi'book) We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ ...
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0answers
13 views

Show that $\exists \beta(z,\bar{z},s)$ such that $\dfrac{1}{2i} [\Lambda, \bar{\Lambda} ]=\beta(z,\bar{z},s)\dfrac{\partial }{\partial s}$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{*}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
2
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1answer
64 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
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0answers
45 views

Divergence Theorem Proof

I am interested in an elementary rigorous proof of say the divergence theorem that is accessible to undergraduates. Avoid differential geometry as much as possible. I know Spivak is one such source ...
2
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3answers
44 views

Manifolds, charts and coordinates

Let's consider the manifold $S^1$ It is well known that we need two charts to cover this manifold. Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle ...
2
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1answer
30 views

Does Heine-Borel hold for smooth manifolds?

If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
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0answers
18 views

Regular Values of a Function

Consider $f(x,y) = x^3 - x + y^2$. What are the regular values of $f$? So I know a point is a critical point if all of the partial derivatives of $f$ are simultaneously zero. So I find that the ...
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1answer
52 views

Definition of a parallelizable manifold

My text that I am self studying from says that a manifold $M$ is parallelizable if it has a trivial tangent bundle which means that there is an isomorphism $\varphi:M\times \mathbb{R}^n\rightarrow ...
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0answers
24 views

Show that $\dim_\Bbb C \mathcal V_p=N-\text{rank}_\Bbb C\left ( \dfrac{\partial \rho}{\partial \overline{Z_j}}(p,\overline{p}) \right )$

I'm learning about Real Submanifolds in $\Bbb C^N$ (from Baouendi's book - page 7): For $p \in \Bbb C^N$ it is customary to denote by $T_{p}^{1,0}\Bbb C^N$ the space of holomorphic tangent vectors ...
1
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1answer
42 views

A Submanifold $M$ of $\Bbb C^N$

I have a Proposition in my book, and I write here: For every $p \in M$, with $M$ be a hypersurface in $\Bbb C^N$ the following hold. \begin{align*} \mathcal V_p &= \left \{ X \in \Bbb C ...
4
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0answers
92 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
1
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2answers
57 views

Explanation of non-orientability of the Möbius band

I have read about the orientation of manifold in the Tu's book. The book is very readable but the first example about non-orientable manifold is seemly hard to understand. On page 208, he gave an ...
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3answers
47 views

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
6
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1answer
77 views

Is this a manifold?

I am trying to get started with differential geometry, and am having a difficult time wrapping my head around the concept of a manifold. One thing that would make it easier to understand would be if ...
2
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1answer
52 views

If $b$ is a regular value of $f$, $f^{-1}(-\infty,b]$ is a regular domain?

I'm trying to prove the first part of Proposition 5.47 of Lee's Smooth Manifolds, which is left to the reader. It says Suppose $M^m$ is a smooth manifold, and $f\colon M\to\mathbb{R}$ smooth. For ...
3
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1answer
87 views

Almost all subgroups of a Lie group are free

I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions. Let $w\colon G\to H$ be an analytic mapping between ...
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0answers
27 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
4
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1answer
69 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
4
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3answers
115 views

Kernel of the Laplacian on a compact manifold

Is there a way to characterise the kernel of the Laplace-Beltrami operator on a compact manifold without boundary? Or is it just "the set of functions $u$ such that $-\Delta u = 0$?"
3
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0answers
35 views

Brief summary of simplicial, CW and manifold notions

I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a ...
2
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1answer
41 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
0
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1answer
32 views

Problem in the proof that $S^n$ is a $n$-dimensional smooth submanifold of $\mathbb R^{n+1}$.

I have the following definition: Definition: $M\subset\mathbb R^{n+k}$ is a $n$-dimensional submanifold of class $C^{p}$ of $\mathbb R^{n+k}$ if for every $x\in M$ there is a neighbourhood $U$ of $x$ ...
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0answers
36 views

differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
2
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1answer
43 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
2
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0answers
34 views

References on the relations between Top, Diff and PL

I have heard many times informal statements like "differentiable and pl manifolds are essentially the same for such and such dimensions", but I would like to know what they mean exactly and how such ...
1
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1answer
45 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
2
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0answers
25 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
0
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1answer
48 views

Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
2
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1answer
46 views

Unit circle can't be covered by one chart

I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
2
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1answer
34 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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2answers
66 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
4
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1answer
73 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
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0answers
45 views

Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ...
2
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1answer
42 views

An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
3
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0answers
45 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...