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

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5
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0answers
78 views

Example of charts on $\mathbb{R}$ that are $\mathcal{C}^r$ compatible but not $\mathcal{C}^{r+1}$ compatible.

Is there a simple example of two charts where this is the case? I'm struggling to think up one.
5
votes
0answers
158 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
0
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0answers
37 views

How do we calculate the Euler numbers of this

Suppose we are given two cubics X(a) and Y(a) in $CP^2$; $X(a)={ (4-a^3) xyz-a^3(x^3+y^3+z^3) =0 }$ $Y(a)={ a(x^3+y^3+z^3)-(2+a^3)xyz =0 }$ where a is a parameter in C satisfying $a^3 \not=1$ and ...
2
votes
1answer
83 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
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0answers
23 views

why f(M) is sub manifold

Let map f of M into N be an injective immersion. show taht if M is compact then f(M) is submanifold of N.
3
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0answers
83 views

Visualizing Frobenius Theorem

Given a smooth vector field $v$ on a (finite dimensional) manifold $M$, one can find the associated integral curves i.e. integral submanifolds of M such that the tangent space at any point $p\in M$ is ...
3
votes
0answers
39 views

Lie differentiation operation for manifolds

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. Let ...
0
votes
1answer
96 views

Intuition of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
1
vote
2answers
104 views

Calculate the Euler Characteristic of M

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$,what is the Euler Characteristic of M?
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1answer
64 views

Is the submanifold compact?

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$, is $M$ compact?
0
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1answer
31 views

Subset of non-units of germs of smooth functions at $x$ is an ideal

For a manifold $X$ with a point $x \in X$ define the ring of germs of the smooth functions at $x$ to be $C^{\infty}_x(X)=C^{\infty}(X)/\sim$ where $f_1 \sim f_2 \iff \exists U \in \tau_X.f_1|U=f_2|U$ ...
6
votes
1answer
142 views

Tangent space for product of submanifolds

Suppose that $X_1$ is an $n_1$-dimensional submanifold of $\mathbb{R}^{N_1}$, and $X_2$ is an $n_2$-dimensional submanifold of $\mathbb{R}^{N_2}$, and let $X=X_1\times X_2$. Let $p_1\in X_1$ and ...
4
votes
1answer
104 views

Product of submanifolds is a submanifold

Suppose that $X_1$ is an $n_1$-dimensional submanifold of $\mathbb{R}^{N_1}$, and $X_2$ is an $n_2$-dimensional submanifold of $\mathbb{R}^{N_2}$. Prove that $X_1\times X_2\subseteq ...
3
votes
1answer
44 views

Derivative of function between sets of matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
7
votes
3answers
279 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
6
votes
1answer
167 views

Proving that Hermitian Metric yields Hermitian Structure on Complex Manifold

Let $g$ be a Riemannian metric on an almost complex manifold $(M,J)$. Suppose $g$ is Hermitian in the sense that $$g(JX,JY) = g(X,Y)$$ Let $\Omega$ be the associated fundamental (Kahler) form ...
1
vote
1answer
105 views

Morse lemma question

Consider the statement of the Morse lemma: Let $b$ be a non-degenerate critical point of $f:M \to \mathbb R$. Then there exists a chart $(x_1, ..., x_n)$ in a neighborhood $U$ of $b$ such that ...
1
vote
1answer
51 views

Why is it that the vector space of all derivations has the basis $\partial/\partial{x_1} \ldots \partial/\partial{x_n}$?

So I have seen stated that two definitions of tangent spaces (w.r.t manifolds) are equivalent. But I am having some difficulty proving they are indeed equivalent. It looks like it boils down to me ...
2
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1answer
348 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 ...
0
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5answers
187 views

Books about manifolds?

I would like to learn about manifolds. Please can someone recommend me a good book to learn about manifolds?
2
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0answers
76 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
2
votes
0answers
72 views

differential of a differential form

Given a differential form $w$ on a manifold, I know how to calculate $dw$ in local coordinates. But is there any way to define $dw$ independent of local coordinates?
4
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2answers
441 views

The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional ...
2
votes
2answers
95 views

Manifold non-orientable iff. frame bundle is connected

Let $M$ be a connected smooth manifold and $L(M):=\bigcup_{x\in M}L_xM$ its frame bundle where $L_xM:=\{(v_1,\dots,v_n):\{v_1,\dots,v_n\}\text{ is a basis of }T_xM\}$. $M$ is non-orientable iff. ...
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vote
2answers
120 views

To find a Coordinate Patch About a Point in Euclidean Subspace.

I have been trying to settle this question for a long time now and it is very important for me to solve this. Let $p, q\in \mathbb R^2$ be points such that $p$ and $q$ are linearly independent (when ...
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vote
0answers
84 views

$SO(2)$ is a manifold?

I am failing to see why $SO(2)$ is a manifold. I see that $SO(2)$ is isomorphic as a group to $([0,2\pi),+_{\mod 2\pi})$. But then when we look for a manifold we need a collection of charts. And since ...
0
votes
1answer
52 views

Extending an embedding $:S^1\rightarrow \mathbb R^{n}$

Assume we have an embedding $f:S^1\rightarrow \mathbb R^n$. I want to extend $f$ to an embedding $\tilde{f}:B\rightarrow \mathbb R^n$, where $B$ is the closed unit ball of $\mathbb R^2$. In fact, I ...
0
votes
0answers
31 views

Existence of vector extensions for the Hessian

Q: $p$ a critical point of a smooth $f$ and $v, w$ two vectors in $T_{p}M$. We extend these two to vector fields $v^{*}, w^{*}$ such that at $p$ the first one equals $v$ and the second one equals $w.$ ...
0
votes
2answers
70 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
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0answers
72 views

Properties of a smooth bijection

What are the basic facts about a map $F: M \to N$ between manifolds (without boundary, we might specify) which is a smooth bijection? The map from $[0,1) \to S^1$ parameterizing the circle is a ...
2
votes
1answer
99 views

[ANSWERED]Lie brackets on vector fields

We consider $v = \frac{\partial}{\partial x}$ and $w = x * \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$. I need to first find the Lie bracket between them which i get to be: ...
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0answers
36 views

levels curves of polynomial equations as manifolds

Q: For which real values c is the subset $f(x) = x_{1}^{2} + x_{1}^{3} - x_{2}^{2} + x_{3}x_{4} = c$ a smooth submanifold of $\mathbb{R}^4$? Try: for it to be a smooth submanifold, $c$ has to be a ...
2
votes
1answer
100 views

Continuous function approximation on manifolds

I am asked to show that every cont. function from a manifold M to $\mathbb{R}$ can be approximated by smooth functions. Try: let f be a map from M to the reals(R). Let ${s_{i}, U_{i}}$ be our atlas. ...
1
vote
1answer
34 views

Is $S^0$ a manifold?

Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does ...
4
votes
0answers
266 views

Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
3
votes
0answers
49 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
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0answers
48 views

Triangulation of a 3-sphere

If one wants to generate a Simplicial complex of the topology of the 3-sphere, one can just take the boundary of a 5-cell, 16-cell or 600-cell. The curvature is concentrated on the edges meeting the ...
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1answer
50 views

Proving orientability of manifold

I don't know how to prove the following: $RP^n$ is orientable manifold if n is odd? Any help is welcome.
0
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1answer
69 views

A Milnor Differential Topology Excercise

If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$. I tried to show that $M^m$ is contractible or convex, but I ...
3
votes
1answer
70 views

Distinction between a vector and a tensor of type (1,0)

Let's say I have a differentiable manifold $\mathscr{M}$. A vector $v$ on this manifold is a map from $\mathscr{F}$ to $\mathbb{R}$, where $\mathscr{F}$ is the set of all smooth functions from ...
1
vote
1answer
90 views

Bump function has a compact support?

Sorry for the basic question, but couldn't find the answer. We say that the bump function $\phi(x)=e^{-1/(1-x^2)}$ has a compact support. However, $\phi(x)\neq 0$ only for $x \in [0,1)$, which means ...
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0answers
85 views

Orientation manifold, what is wrong with my argument?

As I learned, a manifold M is oriented if there exists a smooth nowhere-vanishing n-form on M. So, I am very doubting about the following construction of a n-form $\omega$ on any smooth manifold M (M ...
4
votes
1answer
42 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 ...
1
vote
1answer
27 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
2
votes
2answers
36 views

Can I identify $S_k(V)$ with an homogeneous space?

I'm in trouble with a question: Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Can I identify the manifold, $$S_k(V):=\{(X_1, \ldots, X_k): X_1, \ldots, X_k\in V\ \textrm{are linearly ...
5
votes
1answer
59 views

Proof of 'manifold with dimension less then 4 always has differentiable structure'

L.S., I read in the lecture notes of my course on manifolds (undergraduate) a little side-note that stated that every manifold with dimension less then 4 can be equipped with a differentiable ...
0
votes
0answers
83 views

How to convert limit in polar coordinate system

Let $A=\{(x,y,z):x^2+y^2=z^2,x>0,0<z<1 \}$ oriented by $O$ such that $O^{12}>0$ and $\overline{w}$ be 2-form $\overline{w}=z^2 dy \wedge dz$ then Evaluate:$$\int_{A^0} z^2 dy \wedge dz ...
2
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1answer
99 views

Differentiable manifolds, uniqueness of maximal atlases and definition of smooth manifolds maps.

I have proved that given an atlas for a topological space $M$ that a maximal atlas containing $M$ is unique. But my proof would fail to generalise to the statement that a maximal atlas conatining a ...
0
votes
0answers
60 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
1
vote
1answer
53 views

Tangent spheres to a differentiable manifold

I have the following problem. Let $M$ be a compact $C^r$-manifold (with $r>1$) of dimension $m$ embedded in an euclidean space of dimension $k$. I am told that then there is some $\epsilon >0$ ...