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

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65 views

If $M$ is compact, every maximal ideal in $F$ arises in this way as a point of $M$?

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
7
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1answer
47 views

Tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times DM_2$.

Let $M \subset \mathbb{R}^A$ and $M_2 \subset \mathbb{R}^B$ be smooth manifolds. How do I see that the tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times ...
7
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1answer
135 views

If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite? [duplicate]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is nonorientable and $\partial M$ is empty, then how do I see that $H_1(M, \mathbb{Z})$ is infinite?
7
votes
1answer
50 views

Is the union of an increasing sequence of topological copies of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is ...
7
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1answer
63 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: ...
7
votes
1answer
187 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
7
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1answer
56 views

Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
7
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1answer
100 views

Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
7
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1answer
137 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 ...
7
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2answers
161 views

Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
7
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1answer
479 views

De Rham cohomology of $\mathbb{RP}^{n}$

Consider map from $S^{n}$ to $\mathbb{RP}^{n}$ $$\varphi:S^{n}\to\mathbb{RP}^{n}$$ which maps point $x\in S^{n}$ to corresponding direction in $\mathbb{R}^{n+1}$. This map induces map ...
7
votes
1answer
42 views

If and only if criterion for something to be a differential ideal

Let $I \subset \Omega^*(M)$ be a ($2$-sided) ideal (i.e. $I$ is a vector subspace, and for any $\alpha \in I$ and $\omega \in \Omega^*(M)$ we have $\omega \wedge \alpha \in I$). We say $I$ is a ...
7
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1answer
150 views

Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective? There are space-filling curves, but the usual examples have very "twisty" definitions. UPD A bit of background for this ...
7
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1answer
222 views

If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$.

Question: I am trying to show that if $M$ and $N$ are smooth manifolds (without boundary), and $$F:M\to N$$ is a smooth embedding, then the differential $$dF:TM\to TN,\quad ...
7
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1answer
111 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
7
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2answers
156 views

What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
7
votes
1answer
480 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 ...
7
votes
1answer
124 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let ...
7
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1answer
169 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
7
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1answer
169 views

What is the relationship between Grassmann Manifolds with different dimensions?

I'm an EE student and I'm just beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector ...
7
votes
1answer
418 views

Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
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0answers
96 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
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0answers
129 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal ...
7
votes
1answer
72 views

If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

I am in doubt whether the following statement is true or false: "If M is a manifold of dimension $ n \neq0$ then M has no isolated points." The idea that made me find the true statement was as ...
7
votes
2answers
376 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 ...
6
votes
3answers
624 views

Are all Lie groups Matrix Lie groups?

I have beard a bit about so-called matrix Lie groups. From what I understand (and I don't understand it well) a matrix Lie group is a closed subgroup of $GL_n(\mathbb{C})$. There is also the notion ...
6
votes
5answers
2k 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 ...
6
votes
3answers
705 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). ...
6
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2answers
251 views

Unique factorization of manifolds?

I wonder if there is a result on the unique factorization of manifolds. Call a topological manifold to be indecomposable if it is not homeomorphic to a product of manifolds of positive dimension. Is ...
6
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2answers
2k views

The graph of a smooth real function is a submanifold

Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ which is smooth, show that $$\operatorname{graph}(f) = \{(x,f(x)) \in \mathbb{R}^{n+m} : x \in \mathbb{R}^n\}$$ is a smooth submanifold of ...
6
votes
2answers
82 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
6
votes
2answers
794 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
6
votes
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 ...
6
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2answers
542 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 ...
6
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2answers
376 views

Orientable double covers for non-orientable manifolds

If I have two non-orientable connected manifolds such that their orientable double covers are homeomorphic, can anything be said about the manifolds? Are they homeomorphic?
6
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3answers
264 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
6
votes
2answers
741 views

Manifolds and Charts

I have a very silly and basic question about finding charts for a manifold. The point is: I'm self learning differential geometry, however, I didn't find the answer for this in the book nor on the ...
6
votes
3answers
336 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 $-\Delta u = 0$?"
6
votes
2answers
1k views

First proof of Poincaré Lemma

I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of ...
6
votes
1answer
159 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
6
votes
2answers
322 views

Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
6
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1answer
596 views

How can one prove that manifolds are regular?

First, some clarification of the definition of a manifold that I'm using: A manifold $M$ is a Hausdorff, locally Euclidean and second countable topological space. Now, I am trying to prove that ...
6
votes
3answers
158 views

$2$-dimensional subbundle of tangent bundle of closed $3$-manifold integrable if and only if $\alpha \wedge d\alpha = 0$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. From here and here, I know that there is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any ...
6
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5answers
241 views

How would one define a “manifold” object in prose writing?

I have a question that I fear may raise some objection to the fact that it has been posted here, but I cannot think of a more appropriate place to pose it. I am not a mathematician; I'm a historian, ...
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2answers
1k views

$\mathbb{C}\mathbb{P}^1$ is homeomorphic to $S^2$

I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In ...
6
votes
2answers
272 views

This set is a manifold

let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on ...
6
votes
2answers
127 views

$M$ orientable implies $H_{n-1}(M, \mathbb{Z})$ is free Abelian group. [closed]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is orientable, then $H_{n-1}(M, \mathbb{Z})$ is a free Abelian group?
6
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2answers
202 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
6
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1answer
590 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
6
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2answers
232 views

homomorphisms of $C^{\infty}(\mathbb R^{n})$

Let $F: \mathbb R^{n} \to \mathbb R^{m} $ be a smooth map, then we have homomorphism of algebras $F^{*}: C^{\infty}(\mathbb R^{m}) \to C^{\infty}(\mathbb R^{n})$. Is it true that any homomorphism of ...