# Questions tagged [smooth-manifolds]

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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### Geonetry Manifold

Suppose $M$ and $N$ are $C^r$ manifolds, $r > 1$. Show that $T(M \times N)$ is $C^r>1$
diffeomorphic to $TM \times TN$.

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96 views

### Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...

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### Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...

**5**

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**1**answer

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### Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?

This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)...

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**1**answer

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### Symmetric and anti-symmetric parts of the covariant derivative of a connection

The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program.
Now we come to the question of higher derivatives. As usual in modern
...

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158 views

### Why do unstable manifolds of two close point intersect each other in Baker map?

Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) ...

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### Euler-Lagrange equations on a differentiable manifold

I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...

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**1**answer

278 views

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

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 ...

**3**

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**1**answer

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### Local diffeomorphism on a neighborhood of an embedding

In my reading of the (excellent!) paper of Grabowski and Rotkiewicz on higher vector bundles (https://arxiv.org/abs/math/0702772), I have encountered the following argument which I do not understand. ...

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**1**answer

247 views

### Volume comparison on Grassmannian

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......

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### Uniqueness of Fano varieties

It is a theorem of Kollár–Miyaoka–Mori that there is a finite number of deformation families of smooth, complex Fano $n$-folds for each $n$ (hence also a finite number of diffeomorphism types).
My ...

**2**

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**1**answer

122 views

### Are normal coordinates the same as Cartesian coordinates in flat space?

Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by
$$\exp_p(v)=\gamma_v(1)$$
If we pick any orthonormal ...

**15**

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**1**answer

382 views

### Wu formula for manifolds with boundary

The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...

**1**

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**0**answers

149 views

### Extending Green's theorem from very special regions to more general regions

Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...

**0**

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**1**answer

81 views

### Same fiber of induced covering map [closed]

Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...