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

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1answer
38 views

Is a smooth function still smooth in a sub-manifold containing its image?

Assume $f$ is a smooth map between manifolds $M$ and $N$, and $S$ is a sub-manifold of $N$ with $f(M) \subset S$. Is the map $f: M \to S$ always smooth? Does the answer depend on whether the ...
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0answers
28 views

Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic 3-space. Let $TH$ be the tangent bundle of $H$. I have a question: Is $TH$ isometric to H times a flat $k$-space?
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1answer
59 views

Does every torus $T\subset S^3$ bounds a solid tours $S^1\times D^2\subset S^3$?

I want to show this by using Alexander's Theorem's proof method. So here's what I thought. As I surger $T$, I have 2 $S^2$. So one bounds $S^1$ and the other bounds $D^2$. By reversing the surgery, ...
3
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0answers
57 views

Suggestion for reference book for differential forms, differentiable manifolds and other topics

I am currently taking a course on multivariable calculus and our professor is following the book by Do Carmo: Differential forms and applications. I feel the text is too rigorous, which I really ...
5
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1answer
76 views

Showing that the Mapping Torus is a topological manifold

Let $X$ be a conected topological $n$-manifold, and $f:X\rightarrow X$ an homeomorphism, the Mapping Torus $M_f$ is defined as, $$M_f=X\times [0,1]/\sim$$ where $(x,0)\sim (f(x),1)$. I am trying to ...
3
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1answer
127 views

homeomorphism of the closed disc on $\mathbb R ^2$ [closed]

I just found a statement that for any closed disc on $\mathbb R^2$ and any two points inside it, there is a homeomorphism taking one point to another and is identity on the boundary. But I can't write ...
2
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0answers
29 views

Connect Sum of a connected, compact manifold of dimension n and $S^n$

$M $ be a connected,compact manifold of dimension n. Show that $ M \# S^n$ is homeomorphic to $M$ My idea: $S^n-D^n$ is homeomorphic to $D^n$..so $M \# S^n$ is homeomorphic to $(M-D^n) \cup D^n$ ...
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0answers
130 views

Every Manifold is Locally Path Connected

I'm trying to prove that every topological manifold $M^n$ (with or without boundary) is locally path-connected. My attempt: (Without boundary): Let $x\in M^n$ and $V$ be any open set containing $x$. ...
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0answers
91 views

Find the interior product of a basic p-form $\alpha = dx_1 \wedge dx_2 \wedge \ldots \wedge dx_p$ a and a vector field $X$

I'm reading Differentiable Manifolds by Nigel Hitchin, that is, his class notes for an Oxford course freely available here. In particular, I'm trying to understand the interior product on manifolds, ...
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1answer
48 views

Definition of a vector field on a differentiable manifold.

Wikipedia defines the vector field at a point on a manifold to lie in its tangent space. But is this general enough? Consider a surface traction vector on some manifold, for example. It will have a ...
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0answers
10 views

How to show for every $p\in M$ there is a local chart $(U, x_1, \ldots, x_n)$ around $p$ with desired properties?

Let $M$ be an $n$-dimensional smooth manifold and let $\pi:TM\longrightarrow M$ be the tangent bundle of $M$. Furthermore, let $E$ be a vector subbundle of $TM$ such that $\textrm{dim}(E_p)=k$ for ...
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0answers
34 views

Integrating a $2$ form on $\mathbb C \mathbb P^1$

Assume we have the $2$-form $\omega = \frac 1 {2 \pi i} \frac {dz \wedge d\overline z}{(1 + z \overline z)^2}$ given on $U = \{[1:z] : z \in \mathbb C \} \subset \mathbb C \mathbb P^1$, where $z: U ...
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1answer
22 views

Existence of a smooth map to its embedded submanifold

Assume $M$ is a manifold, and $N$ is an embedded submanifold of $M$. Is there always a smooth map $f$ from $M$ to $N$, s.t. $f|_N=\text{id}_N$?
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0answers
29 views

smooth Banach manifold structure on C^1(M,N), smoothnes of transtion maps.

Let $M$ and $N$ be finite dimensional compact manifolds. Denote the set of $C^1$-maps from $M$ to $N$ by $C^1(M,N)$. To my knowledge, the following is true: If we equip $C^1(M,N)$ with the ...
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0answers
20 views

A question about $[X,Y]$, where $X,Y$ are vector fields

Here we set $X,Y\in \Gamma(TM)$, $f\in C^\infty(M)$, $q\in M$. Then when we consider $[X,Y]$ as the derivation, what is $[X,Y](f)(q)$? Is that $X_qY_q(f)-Y_qX_q(f)$?
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1answer
46 views

Is it true that a geodesic on a hyperbolic surface can be lifted to a geodesic on the hyperbolic plane?

Let $\mathbb{H}$ be the hyperbolic plane, $\Gamma < \text{Isom}(\mathbb{H})$ be a Fuchsian group, and $S = \mathbb{H}/\Gamma$. If $\gamma : [0,1] \rightarrow S$ is a geodesic, can it be lifted to a ...
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2answers
49 views

Difficulty comprehending this sentence on Wikipedia

Via Wikipedia https://en.wikipedia.org/wiki/Exterior_algebra#Differential_geometry, there is this given definition "In particular, the exterior derivative gives the exterior algebra of differential ...
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2answers
39 views

Manifold structure of $\mathrm{PSL}(2,\mathbb{R})$

I am trying to prove that $\mathrm{PSL}(2,\mathbb{R})$ is a 3-dimensional connected non-compact Lie group, where $$\mathrm{PSL}(2,\mathbb{R}) = ...
2
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0answers
55 views

Existence of a free “sub-isometric” embedding of a Riemannian manifold

I am trying to understand Matthias Günther's proof of Nash's embedding theorem which is outlined in Günther, Matthias. Isometric embeddings of Riemannian manifolds. Proceedings of the International ...
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1answer
48 views

Example of smooth function of a smooth submanifold cannot be obtained by restriction of a smooth function in the manifold

Anyone can give the example of smooth function of a smooth submanifold N of M cannot be obtained by restriction of a smooth function in the smooth manifold M?
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2answers
73 views

How to prove $O(n,C) $ is not compact

How does one prove that $O(n,C) $ is not compact? I am guessing it can be done by showing it is not bounded.
1
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1answer
79 views

Tangent vectors as derivatives

I am reading "An introduction to manifolds" by Loring Tu and I struggle understanding a concept. In the book the directional derivative is given a tangent vector $v$ at a point $p$ the map : $$D_v : ...
8
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1answer
64 views

Can intersection of two manifolds be $xy=0$

I know that if two manifolds intersect transversally then their intersection is a manifold. But I was trying to construct an example where the intersection is not a manifold. But I still do not see ...
3
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0answers
22 views

Vanishing of index of elliptic operators on odd dimensional manifolds

It is known that if $D$ is an elliptic differential operator on $M$ which is assumed to be odd dimensional, then index o $D$ vanishes. It essentially follows from the index formula in cohomology ...
2
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1answer
66 views

Integral of the Laplace-Beltrami Operator multiplied by a function

I have the following problem: Let $\mathcal{M}$ be a $2D$-manifold in $\mathbb{R}^3$ and let $g$ denote its metric. Furthermore it is known that $\mathcal{M}$ is a closed manifold (i.e. it has no ...
4
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1answer
99 views

Struggling to understand real projective space

My ultimate goal is to show that the real projective space $\mathbb{P}^n_{\mathbb{R}}$ is an $n$-manifold. But first I'd like to understand the topological structure of $\mathbb{P}^n_{\mathbb{R}}$. ...
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0answers
19 views

Y is $f_t$-related to Y implies $f_t(g_s(p)=g_s(f_t(p))$ for all p$\in M$?

I saw this claim, but stuck on this for a long time: $X,Y\in \Gamma(TM$) are two complete vector fields with flows {$f_t$},{$g_s$}, then Y is $f_t$-related to Y implies $f_t(g_s(p)=g_s(f_t(p))$ for ...
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0answers
17 views

The integral curve of the points around critical point of the vector field

Suppose to vector field V, $a$ is its critical point, how to show there exits a NBHD V s.t. integral curve exists on [-1,1] for all the points from V?
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0answers
81 views

Showing diffeomorphism using partition of unity

How do I show there exists a diffeomorphism $f: [0,1) \rightarrow [0, \infty)$ such that $f(x) = x$ for small $x$ using partition of unity?
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0answers
44 views

exactness of product of forms

Given two forms α and β, let α be closed and β be exact, how do you prove that αβ is exact? I can see that αβ is closed, but that is only sufficient fact from the product being exact. Any suggestions? ...
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0answers
29 views

Question regarding the dimension of a manifold of zero locus of a cubic homognenous polynomial

Let $f \in \mathbb{Q}[x_1, ..., x_n]$ be a homogeneous polynomial of degree $3$. In an article I am reading, they claim that the zero locus of $f$ over $\mathbb{R}$ inside the interior of $[-1,1]^n$ ...
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1answer
79 views

Does the zero locus of homogeneous polynomial always form a manifold?

These are possibly very basic questions. My knowledge in manifolds is quite limited and I would appreciate any comments! Suppose I have $f \in \mathbb{Q}[x_1, ..., x_n]$ that is homogeneous of ...
4
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0answers
19 views

$\theta_{(U, \phi, p)}(\vec{u}) = [(U, \phi, \vec{u})]$ a bijection?

Let $M$ be a $C^k$ manifold of dimension $n$, where $k \ge 1$. Let $(U, \phi)$ be a chart for $M$ around a point $p \in M$, and let $\theta_{(U, \phi, p)}: \mathbb{R}^n \to T_pM$ be defined as ...
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1answer
44 views

Degree of $P$ as a smooth function equals its polynomial degree $d$.

Let $M,N$ be connected oriented manifolds such that $\partial M=\partial N=\emptyset$. Let $F:M\to N $ be a smooth proper map (i.e. for every $K\subset N$ compact, $F^{-1}(K)$ is compact). We define ...
2
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1answer
17 views

What is the relationship between the Liouvelle measure on $T^1(\mathbb{H})$, and the Haar measure $\mu$ on $PSL(2, \mathbb{R})$?

Let $\frac{dx dx d\theta}{y^2}$ be the Liouvelle measure on $T^1(\mathbb{H}) \cong \mathbb{H} \times \mathbb{S}^1$. Let $\mu$ be the Haar measure on $PSL(2, \mathbb{R})$. What is the relationship ...
3
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1answer
121 views

Homology of non-orientable manifolds.

Is there any example of non-orientable manifold with boundary M of dimension n>2 with the property that $H_i(M, Z_2)$ vanishes for all $0<i<n$? If it does not exist, is there any construction ...
2
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1answer
84 views

Two non-equivalent atlas on $\mathbb S^1$

In the same way that I can find two non-equivalent atlas for ${\mathbb{R}}$ (in fact I found an infinite countable number of them) I'm trying to find two non-equivalent atlas for ${\mathbb{S^1}}$. I ...
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0answers
31 views

Intuition of transversality equation

I am studying Differential topology from Guillemin / Pollack and unfortunately i cannot understand ıntuition of Transversality equation Let $f$ be a smooth map between smooth manifolds $X$ and $Y$ ...
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2answers
129 views

Why aren't two cones attached in their vertex a manifold? [closed]

In this general relativity lecture notes, on page two, it is claimed that two cones attached on their vertex do not constitute a manifold. I don't see why. So, why?
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1answer
49 views
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1answer
67 views

Two problems about manifolds [closed]

1.$M$ is a connected manifold, dim$M \geq 2$, $f:M \rightarrow \mathbb{R}$ is smooth, then $f$ is not an injection. 2.$M, N$ are two manifolds, and $M$ is connected, $f:M \rightarrow N $ is smooth,if ...
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0answers
24 views

Two segments Manifold

Let consider the set composed by two intersecting segments of $R^2$. Is it (sub)-manifold ? Thanks
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1answer
41 views

Pulling back forms computation

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function mapping $$(x, y) \to \left(e^{2x}, xy\right).$$ How do you compute pulling back form of a 2-form $$\alpha(x, y)=xy(dxdy),$$ in other words, ...
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0answers
45 views

Calculating Equilibria for a 3 ODE system with 11 unknown parameters i.e. Looking For the intersection points of 3 surfaces.

I have the following system of 3 ODEs of 3 variables $(\ell,M,h)$ and 11 parameters $(\sigma_{\ell},\mu_{\ell},d_{\ell},\sigma_M,\alpha_1,\beta,\alpha_2,\nu_M,\sigma_h,\nu_h,d_h)$: ...
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0answers
43 views

Domain of integration

I am reading through Lee's Introduction to smooth manifolds and he defines a domain of integration to be a bounded subset of $\mathbb{R}^n$ whose boundary has measure zero. He then goes on to say ...
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1answer
29 views

prove that $n$-manifold has a countable basic

$M$ is a $n$-topological manifold if it is a a topology space and each point of $M$ has a neighborhood that is homeomorphic to $\mathbb{R}^n$. How do I prove that $n$-topological manifold has a ...
5
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0answers
115 views

Connected topological manifolds

For any connected topological manifold, it is true that for any two points on the manifold, there exists a single local chart that both of two points lie in it. How can I prove it?
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1answer
41 views

Deriving formula for formal adjoint

My question is in relation to the derivation of the formal adjoint for a connection $D:\Omega^{p-1}(\text{Ad}E)\rightarrow \Omega^p(\text{Ad}E)$ - I am reading through this derivation in Jost's ...
0
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1answer
23 views

Calculating intersection numbers of surfaces in example from knot surgy

There is a section in an explanation about knot surgery which I do not understand in "Knot surgery revisited" by Fintushel, p.203. Let $X$ be a simply-connected compact $4$-manifold. Let $K$ be a ...
2
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1answer
75 views

Two different approaches to defining Lie algebra of a Lie group

I've been reading two books that touch on the Lie theory: Representation Theory by Fulton & Harris and Introduction to Manifolds by Loring Tu. They define Lie bracket on tangent space at identity ...