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

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7 views

Real/Complex Manifolds - Transition Maps

I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. I can read the definitions and work with them, but I want to understand ...
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2answers
30 views

connected sum of two surfaces

I was reading Massey's textbook on Algebraic topology and the author claims that if $S_2$ is a 2-sphere then $S_1 \# S_2$ is homeomorphic to $S_1$. I don't know why that is true and since I'm very ...
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1answer
35 views

Braid Groups on Manifolds

I am studying braid groups on manifolds and am getting confused. In a geometric definition, one needs to first choose a simple curve $\theta$ on a given manifold $M$ and well-ordered points ...
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1answer
35 views

Codimension 1 homology represented by Embedded Submanifold

I'm looking for a reference for the following statement: Given an oriented manifold $M$ and a class $\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold $F^+ \hookrightarrow M$ such that ...
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4answers
743 views

Why 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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1answer
24 views

Knots as boundaries

Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process. In 2D space, a simple closed curve in a plane can be thought of as a boundary of ...
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3answers
54 views

What exactly is a 0-form?

From what I understand, a k-form in the real numbers is essentially a mapping $\mathbb{R^k} \rightarrow \mathbb{R}$, but I can't seem to find a corresponding definition for a "0-form". Wikipedia seems ...
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1answer
33 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
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0answers
16 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
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1answer
20 views

Geometric translation of a theorem about stability of equilibrium point

In the book Nonlinear Systems by Hassan Khalil, there is a theorem about the stability of equilibrium point ‎‎ which asserts that : Theorem :‎ Let‎ ‎$X = 0 $ ‎be an equilibrium point for‎ ‎$‎\dot{x} ...
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1answer
29 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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1answer
40 views

Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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1answer
35 views

trace map is continuous

Prove that $tr: M_n(k)\to k$ is continuous. I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good. ...
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0answers
17 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...
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0answers
31 views

Compact manifold with smooth structure

Is the following surface smooth and compact, when all its partial derivatives are continuous? How to tell about self-intersections without visualization? $x=\cos(a) + \cos(a + b) + \cos(a + b + ...
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0answers
28 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
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1answer
41 views

Reference request: Measure theory and/or manifolds [duplicate]

I have never encountered measure theory or manifolds yet, despite being close to my third year university level. Any texts for either or both of these subjects would be greatly appreciated.
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1answer
21 views

Identity concerning push forward of two vector fields

How would you prove the identity $\displaystyle \frac{\partial}{\partial s}\Psi_{s^*} \mathbb{X} = (-1)L_{\mathbb{Y}}\Psi_{s^*}\mathbb{X}$ where $\Psi_{s}$ is the flow of $\mathbb{Y}$ and ...
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1answer
37 views

tangent bundle and normal bundle

I have a problem about tangent bundle. It is known that the tangent bundle of most manifolds is not trivial: for example, the tangent bundle for $S^2$ is not $S^2\times \mathbb{R}^2$. However, for a ...
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0answers
14 views

Show that $M$ is not equivalent to $O_k$.

Assume that a model hypersurfaces is described by $$O_k=\{(z,w)\in \Bbb C^2 \mid v=|z|^k\}\tag 1$$ and a real analytic hypersurface: $$M=\left \{(z,w)\in \Bbb C^2 \mid ...
8
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1answer
107 views

Why are Banach manifolds not so popular?

Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the ...
4
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1answer
35 views

Converse to the Jordan-Brouwer separation theorem

By the Jordan curve theorem, if $C \subset S^2$ is (the image of) a simple closed curve, then $S^2 \setminus C$ has precisely two connected components. This statement admits the following "converse". ...
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0answers
97 views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
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1answer
14 views

Writing nonautonomous systems as autonomous systems

Apparently any mth order nonautonomous system is equivalent to a first order autonomous system in higher dimensional space. How does this work in practice? I would you write $\displaystyle ...
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1answer
30 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
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3answers
50 views

A proof of compactness, connectedness of real projective space

I need a reference for a complete proof of the below theorem: Let $RP^n$ be $n$-dimensional real projective space. Then $RP^n$ is a compact, connected manifold. (Consider the standard topology over ...
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1answer
14 views

Equivalence relation of differential forms

My notes claim that $\displaystyle d\omega (x) = \frac{1}{k!} d\omega_{i_1\cdots i_k} \wedge f^{(i_1)}\wedge\cdots\wedge f^{(i_k)}$ is equivalent to $\displaystyle d\omega(x) = \frac{1}{k!} ...
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1answer
38 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
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1answer
40 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
0
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2answers
43 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
66 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 ...
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3answers
57 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
54 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 ...
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1answer
98 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
31 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
39 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) ...
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1answer
56 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
101 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 ...
2
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1answer
39 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
37 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 ...
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1answer
42 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 ...
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0answers
64 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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1answer
77 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
47 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 ...
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3answers
48 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
33 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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1answer
56 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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1answer
46 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 ...
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0answers
95 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 ...