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

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Gluing manifolds

Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this: A point $(\cos \phi , \sin \phi, ...
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141 views

Does every non-compact manifold admit an incomplete vector field?

I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess ...
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151 views

special covering of a non-compact manifold

I'm very stuck on the following exercise in the book "A Comprehensive Introduction to Differential Geometry V.1" by Michael Spivak: Let $M^m$ be a smooth connected non-compact manifold. Show that $M$ ...
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298 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
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Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M ...
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Heegaard Splitting of Brieskorn homology 3-spheres

For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. I want to know ...
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135 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
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Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
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380 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 ...
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vector space of all smooth functions has infinite dimension

Now, I am working through a particular case in the book on smooth manifolds by John.M.Lee used in my graduate math class, let's say we have a smooth manifold X which has positive dimension. He then ...
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512 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 ...
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78 views

Does $\mathrm{Mat}_{m \times n}$ have boundary?

To me, $\mathrm{Mat}_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: An exercise on Regular Value Theorem. ...
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225 views

How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
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215 views

Classifying vector bundles

Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
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1answer
117 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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244 views

map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...
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76 views

Understanding the proof for: $d(f^*\omega)\overset{!}{=}f^*(d\omega)$

Consider this Proposition: Let $U\subset\mathbb{R}^n$ and $V\subset\mathbb{R}^n$ be open sets and $\phi:U\to V$ be differentiable. For all $k\in\mathbb{N}_0$ and $\omega\in \Lambda^k(V)$ it is true ...
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157 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
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2answers
206 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
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327 views

Topology on Klein bottle?

I was trying to show that the Klein bottle was second countable. My try was to use that it has the subspace topology of $\mathbb R^3$. Then I noticed that it is not imbeddable into $\mathbb R^3$. ...
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1answer
240 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
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1answer
576 views

Manifold embedded in euclidean space with nontrivial normal bundle

Let $X$ be a differentiable $n$-manifold embedded in some $\mathbb{R}^{n+1}$. I have two questions. I have read that if $X$ is compact and orientable, then the normal bundle of the embedding is ...
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62 views

Why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$?

Given a real $m$-dimensional smooth manifold why do we consider tangent spaces and what not, when we can just use Whiteney's Embedding Theorem and do calculus in $\mathbb{R^{2m}}$? I assume there is ...
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281 views

Pushforward of Lie Bracket

I am trying to figure out why the following equality is true : $$f_*[X,Y]=[f_*X,f_*Y]$$ where $f:M\rightarrow N$ is a diffeomorphism, $M$, $N$ are smooth manifolds, $X$, $Y$ are smooth vector fields ...
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1answer
341 views

How to prove figure eight is not a manifold? [duplicate]

Possible Duplicate: A wedge sum of circles without the gluing point is not path connected I know that figure eight is not a manifold because its center has no neighborhood homeomorphic to ...
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2answers
493 views

Manifold with different differential structure but diffeomorphic

I'm new to differential geometry and reading Lee's book Manifold and Differential Geometry. In the first chapter, he mentioned the following two maps on $\mathbb{R}^n$: (1) $id: (x_1,x_2\cdots x_n) ...
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1answer
69 views

Does a germ of a smooth (i.e., $C^\infty$) function at a point of a manifold always extend to a global smooth function?

Obviously this doesn't hold if we replace "smooth" with something like "analytic" or "regular," which are the contexts I'm more familiar with. And obviously we can't extend a smooth function defined ...
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1answer
53 views

How to show $f^*E$ is a smooth submanifold…

I'm wondering how to show the following: let $E$, $B_1$ and $B_2$ smooth manifolds. Suppose $\rho:E\rightarrow B_2$ is a smooth vector bundle and $f:B_1\rightarrow B_2$ a smooth map. If we write ...
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1answer
101 views

Is there a name for this particular class of topological space?

This is a simple question, but I can't figure out the name for this class of topological space. Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric. Now, say ...
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87 views

Basic property of a tensor

In Jost's Riemannian Geometry and Geometric Analysis (6th ed.) on page 142 there is the following remark concerning the torsion tensor. Remark. It is not difficult to verify that [the torsion ...
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1answer
63 views

Geodesic complete subset of a connected manifold

This may be a very silly question but let us consider a connected Riemanian manifold $(M,g)$ and a subset $O\subset M$. Can we have $O$ geodesic complete (in the sense of all geodesics linking two ...
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110 views

Differentiable manifolds $\mathscr C^k$ vs. $\mathscr C^\infty$

I noticed that there exists a (in some sense) better definition of the tangent space via the dual of a certain quotient algebra which is easier to work with in some cases. This however only works for ...
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1answer
101 views

orbits are open in Manifold ? group action on manifold.

I need to show: for a differentiable manifold $M$, and $Aut(M)$ acts on $M$, orbit of a point $a\in M$ is open in $M$, please help.
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310 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 ...
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490 views

De Rham cohomology of $S^2\setminus \{k~\text{points}\}$

Am I right that de Rham cohomology $H^k(S^2\setminus \{k~\text{points}\})$ of $2-$dimensional sphere without $k$ points are $$H^0 = \mathbb{R}$$ $$H^2 = \mathbb{R}^{N}$$ $$H^1 = \mathbb{R}^{N+k-1}?$$ ...
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150 views

Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
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125 views

Most important aspects of differential geometry for general relativity

I'm an undergraduate getting ready to take a graduate course in general relativity next quarter. I purchased Wald's General Relativity (who incidentally will be teaching the class) in order to get a ...
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1answer
40 views

Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that ...
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159 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
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1answer
250 views

$M/\Gamma$ is orientable iff the elements of $\Gamma$ are orientation-preserving

The probem is: Suppose $M$ is connected, orientable, smooth manifold and $\Gamma$ is a discrete group acting smoothly, freely, and properly on $M$. We say that the action is ...
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1answer
140 views

$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
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238 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
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1answer
117 views

Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?

I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds: Example 9.3. Let $\Gamma$ be the graph of the function $f(x) = \sin(1/x)$ on the ...
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1answer
117 views

Non-vanishing vector field on $\mathbb{R}P^{2n+1}$

I'm trying to cook up a non-vanishing vector field on $\mathbb{R}P^{2n+1}$. I know that $S^{2n+1}$ admits one, namely $(x_1,\dots,x_{2n+2})\mapsto (-x_2,x_1,\dots,-x_{2n+2},x_{2n+1})$. Moreover, I ...
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2answers
161 views

Proving homeomorphism between surface and $\mathbb{R}^2$ minus Cantor Set

I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - ...
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1answer
99 views

Proving that the smooth automorphism group of a manifold $M$ acts transitively on $M$

Let $M$ be a differentiable manifold of dimension $n$, let $p,q\in M$ any two points. We need to show there exists an automorphism $f\in \mathrm{Diff}(M)$ with the property that $f(p)=q$. Could ...
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1answer
106 views

$U(n)/U(n-1)$ as homogeneous space

How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
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113 views

elementary question regarding differential forms

Is it possible to give a high level explanation why changing the order of differentials will give rise to a minus sign ? I.e. why do we have $$ dx\,dt = - dt\,dx $$ (I am going to take a course on ...
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1answer
165 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
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1answer
87 views

metric on the Euclidean Group

I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for ...