For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
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57 views
Tangent bundle : is a manifold
I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
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1answer
224 views
Uniformization Theorem for compact surface
Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
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1answer
363 views
Relationship Between Differential Forms and Vector Fields
I am trying reach an understanding of precisely how the space of differential forms is related to
the space of vector fields. These are the definitions that I understand and am using for these ...
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1answer
144 views
The embedding of smooth manifold
I have run into a problem in my differential geometry book.
Let $M$ be a smooth manifold and $F={C^\infty }(M,\mathbb R)$. Define a mapping $i:M \to {\mathbb R^F}$ by ${i_f}(x) = f(x)$ for $x \in M,f ...
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2answers
96 views
Inducing orientations on boundary manifolds
Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
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67 views
Topological space M with partition of unity--->M paracompact. John Lee Problems
Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
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1answer
53 views
Embedding manifolds of constant curvature in manifolds of other curvatures
I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
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1answer
51 views
Finding an algebra of smooth functions on a manifold with a given product.
I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
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138 views
Are matrices with determinant zero a manifold?
Consider the set of matrices with determinant zero in $M_n(\mathbb R)$, where $n > 1$. Is it a manifold? In fact, is it even a topological manifold?
I would suspect not; but I do not have a proof.
...
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1answer
127 views
Constant Rank theorem for domain with nonempty boundary
Problem 4-3 in J.M. Lee's introductory text about smooth manifolds, asks to formulate and prove a version of the constant rank theorem for a map of constant rank whose domain is a smooth manifold with ...
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1answer
233 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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1answer
87 views
Diffeomorphic riemannian manifolds and volume forms
Maybe the question will be stupid, but I'm a beginner in riemannian geometry...
We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
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1answer
71 views
Isometries from Diffeomorphisms
Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?
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1answer
107 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 ...
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0answers
57 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 ...
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0answers
79 views
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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0answers
88 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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1answer
99 views
Status of PL topology
I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
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0answers
114 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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2k views
Good introductory book on Calculus on Manifolds
I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq.
I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject?
Should I ...
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votes
2answers
319 views
Intuition for not-so-smooth manifolds
in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds.
However, neither in ...
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3answers
999 views
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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2answers
1k views
Manifold Explained
Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one.
My ...
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2answers
221 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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2answers
84 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
97 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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3answers
266 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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2answers
123 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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votes
2answers
78 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
71 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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2answers
347 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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2answers
140 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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votes
2answers
94 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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2answers
145 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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1answer
113 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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2answers
58 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
68 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
71 views
The open Möbius Band is not orientable
Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
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1answer
58 views
Differential of smooth function on manifold
In the book I am using, the author defines differentials in the following way.
Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
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1answer
55 views
Vector Bundle Doubt..
Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
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1answer
107 views
Why do we need Lie derivative?
If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative?
In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
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2answers
60 views
Closed ball not a manifold.
My book on differential geometry claims that a closed ball in $\Bbb R^n$ can never be a differentiable manifold because of the boundary points. The book doesn't really give an explanation for why this ...
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3answers
138 views
Physics notation justified
Sometimes in physics they do things like this one:
If $dq=f\left(x\right)\cdot dr$ then $\frac{dq}{dt}=f\left(x\right)\cdot \frac{dr}{dt}$
Which mathematically is a wrong deduction. Is there any ...
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1answer
71 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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2answers
128 views
Connected manifold + hole = connected manifold?
If I have a connected manifold and I poke a hole in the interior of the manifold, it seems obvious that the manifold is still connected. But how would you prove this?
More precisely, let $M$ be a ...
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1answer
165 views
How to show that “spheres” are diffeomorphic?
Can anyone provide a diffeomorphism between these "spheres": $\mathbb{S}^2$ and $\{(x,y,z)\in \mathbb{R}^3: x^4+y^2+z^2=1\}$?
PS: If you know a result that can solve this problem, I would be glad to ...
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1answer
146 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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2answers
48 views
The zero set of $z_0^2+z_1^2-1$ in $\mathbb{C}^2$.
Recently, I read the notes "Vector bundles on Riemann surfaces" by Sabin Cautis (http://www-bcf.usc.edu/~cautis/classes/notes-bundles.pdf). On the sixth page of these notes, there is a statement ...
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1answer
149 views
Different definitions of handle attachment
This is an extremely technical question about handle attachments. I am asking why two definitions are equivalent. My question appears in the second to last paragraph after I've described the two ...
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1answer
117 views
Green's function for the Yamabe problem
I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8.
Theorem 2.8 (Existence of the Green Function).
Suppose $M$ is a ...
