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

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55 views

Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) ...
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2answers
2k views

Orientability of projective space

Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even. First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
7
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1answer
666 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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2answers
2k views

Why are Riemann surfaces algebraic curves?

I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N--I know how the Riemann surface is ...
7
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2answers
194 views

Do we know this homogeneous space by another name?

Consider the homogeneous space $GL(3)/GL(2) = GL(3,\mathbb{R})/GL(2,\mathbb{R})$ where $GL(2)$ fixes the first coordinate axis (so can be identified with the subgroup of $2\times 2$ blocks sitting in ...
7
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1answer
903 views

Converse To Quotient Manifold Theorem [Exercise in Lee Smooth Manifolds]

I would like help with the following problem (chapter 9, #4) from Lee's Smooth Manifolds [its not homework, I'm reading it and I got stuck on this one] If a Lie group $G$ acts smoothly and freely on ...
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2answers
69 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 \...
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1answer
49 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
140 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?
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1answer
51 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 ...
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1answer
68 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
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1answer
204 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 $\nu_{\...
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1answer
57 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
138 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 ...
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2answers
165 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 $...
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1answer
508 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 $${\varphi}^{*}:\...
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1answer
44 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
153 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
229 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 dF(p,v)=(F(p),dF_p(v))...
7
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1answer
123 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 \pi_n(...
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2answers
168 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
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1answer
488 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
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1answer
126 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 $n=dim(...
7
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1answer
174 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
177 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
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1answer
421 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 $v\in T_pM$ $$\frac{D}{dt}\bigg|_{t=0}(d\...
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0answers
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Sard's Theorem Using Integration?

How exactly do we clean up this heuristic proof of Sard's theorem, from Schwarz' `Differential Topology for Physicists' using the Jacobian $J(f)(x)$: "A heuristic justification for Sard's theorem ...
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Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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113 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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187 views

Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
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1answer
75 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 ...
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0answers
103 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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143 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
7
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2answers
402 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
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3answers
663 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 ...
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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 ...
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3answers
720 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
816 views

Do diffeomorphisms act transitively on a manifold?

Let $M$ be a smooth manifold, $x,y\in M$. Must there exist a diffeomorphism $f : M \rightarrow M$ with $f(x) = y$? I tried proving this via vector fields, i.e. trying to find a vector field whose ...
6
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2answers
256 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 ...
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2answers
863 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 ...
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2answers
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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 $...
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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 ...
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2answers
552 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 ...
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2answers
389 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
269 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 $\mathbb{C}P^{1}\simeq S^{2}$ ...
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2answers
757 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 web....
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1answer
176 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 M)\...
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3answers
361 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
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2answers
987 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...