# Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

2,942 questions
3answers
25 views

### How to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, (x^1, …, x^n))$ is a local chart of a smooth manifold.

Let $M$ be a smooth manifold. I am getting lost in the notations. Could someone please explain me how to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, \phi = (x^1, ..., x^n))$ is a local ...
0answers
20 views

0answers
39 views

0answers
53 views

0answers
82 views

### Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$? [migrated]

This question was prompted by my answer to this question. An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard ...
1answer
43 views

1answer
27 views

1answer
42 views

### Prove that the graph of a function is a manifold

I know how to show this if $X$ and $Y$ are euclidean spaces using IFT but wanted to confirm proofs about the abstract case. Q) a) $X$, $Y$ are smooth manifolds and $f:X\rightarrow Y$ is smooth. Show ...
2answers
138 views

### How to prove that $f(x)=\frac{x}{\sqrt{1+|x|^2}}$ is smooth?

I have a function $f(x)=\frac{x}{\sqrt{1+|x|^2}}$ on the open ball $B_1(0)$ in $\mathbb{R}^n$. I have to show that this function is a diffeomorphism between $\mathbb{R}^n$ and $B_1(0)$. I have already ...
1answer
40 views

### Derivative of products of exponential maps

Let $G$ be a (finite-dimensional) Lie group with Lie algebra $\mathfrak g$. Then for $f\in C^\infty(G)$, $X,Y\in\mathfrak g$ and $g\in G$ one can define $f(ge^{tX})\in C^\infty(\mathbb R)$ which ...
0answers
17 views

1answer
32 views

### Existence of degree of smooth map between manifold and sphere

I came across this statement and couldn't figure out why this is true, please help: Let $M$ be an n-dimensional compact, connected, orientable smooth manifold without boundary. Prove that there ...
1answer
54 views

### Rational Euler number of oriented Seifert manifolds

I'm studying Michele Audin's book - Torus Actions on Symplectic Manifolds and stumbled across an exercise I can't prove. Exercise I.13 Prove that the Euler class of the Seifert manifold with ...
0answers
32 views

1answer
23 views

0answers
43 views

### Why can't we define any $C^{\infty}$ structure on the single point $0 \in \Bbb R^n$?

I'm studying the book "Differentiable Manifolds" by Brickell &Clark . On page 94, in the example 6.3.6, it was written that we can't define any $C^{\infty}$ structure on the single point $0$. ...
1answer
88 views

### Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
0answers
39 views

### Is this function an open mapping? $(X,Y)\mapsto Yh(p)\cdot X(p) - Xh(p) Y(p).$

Let $M$ be a compact smooth $3$-manifold, and $h: M\to \mathbb{R}$ a function such that $\{0\}$ is a regular value of $h$, and define $\Sigma = h^{-1}(0).$ Moreover, we will denote $\mathfrak{X}^r(M)$ ...
1answer
58 views

### Chern classes, classification of bundles, and Bockstein morphism

I'm doing a work on Chern classes and I have the following doubts, I do not know if anyone could support me with the doubts, or with bibliography / references Given a $n \in \mathbb{Z}$, is it ...
0answers
33 views

### Differential of the inverse map gives rise to the bracket of vector fields being zero

Suppose that $J:G\to G$ is the inverse map on a Lie group $G$ And suppose that $X,Y$ are vector fields. We know that if $\phi:G\to G$ is an arbitrary diffeomorphism then $d\phi[X,Y]=[d\phi X, d\phi Y]$...