# Questions tagged [smooth-manifolds]

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

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### Smoothness of a homomorphism on Lie subgroup

Suppose $\phi:G_1\to G_2$ is a Lie group homomorphism where $G_1$ is connected. Let $\phi(G_1)\subseteq G_3$ and $G_3$ is a Lie subgroup of $G_2.$ I want to prove that $\phi:G_1\to G_3$ is a Lie group ...
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### How to obtain real vector from abstract tangent vector in the case of the manifold $\mathbb R^n$

I know that for every $p\in\mathbb{R}^n$ the map \begin{align} \Phi_p\colon\mathbb{R}^n&\to T_p\mathbb{R}^n\\ v&\mapsto D_{v,p} \end{align} is an isomorphism, where \begin{align} D_{v,p}\...
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### Wave map on manifolds

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold. Now in a paper ...
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### Show that $C^\infty(\overline{\Omega}) \subseteq C^{0,1} (\overline{\Omega})$

Let $\Omega$ be a bounded, connected, open domain in $\mathbb{R}^d$ with smooth boundary. Denote by $C^{0,1} (\overline{\Omega})$ the space of continuous functions $u$ on $\overline{\Omega}$ such ...
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### Structure preserving maps between manifolds with boundary

While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere. Namely I want to ...
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### Confusion about the Definition of Smooth Functions on a Manifold

I am slightly confused about the definition of smooth functions on a smooth manifold given in An Introduction to Manifolds by Loring Tu (Second Edition, page no. 59). The definition is given below. I ...
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### Interesting examples of submersions that are not surjective

What are some "interesting" examples of submersions that are not surjective? Usually, the notion of submersion comes with a prefix of surjective. Most of the maps we come across when we do ...
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### Is there a definition of limiting flatness on a manifold?

I am not talking about local flatness. Local flatness means an entire chart can be mapped to a flat euclidean space. For this reason for example a sphere is not flat, because any chart will still be ...
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### Hint in an exercise, manifold and its dimension

I'm triynig to solve this exercicse but I have a trouble: I don't know what is the dimension of $G$, it is finite or not? but I suppose the base is $(x_1,...x_n)$ so I can wrote any vector as follows ...
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### Connections and second derivatives of curves

If $M$ is a smooth manifold, let $AM$ be the subspace of the double tangent bundle $TTM$ consisting of vectors $v$ such that $\pi^{TM}(v) = \pi^X_*(v)$, where $\pi^X: TX \to X$ is the projection of a ...
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### construct nonvanishing one form on $\mathbb{RP}^3$

I met such a problem in my homework: construct an everywhere nonvanishing 1-form on $\mathbb{RP}^3$; can your construction be generalized to $\mathbb{RP}^n$? I know nothing about the background of ...
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### Binary representation unusual relation in theorem about immersed manifolds.

Question I'm reading Schuller's Lectures on the Geometric Anatomy of Theoretical Physics and he states the following theorem. I was surprised by this theorem and would like references for further ...
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### Wrong proof of $TM$ is diffeomorphic to $M\times \mathbb{R^m}$

I want to see what's wrong in here: Let $M$ be a smooth manifold with dimension $m$. I will show $TM$ is diffeomorphic to $M\times \mathbb{R^m}$. proof) Define $F:TM\rightarrow M\times \mathbb{R^m}$ ...
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### Tangent space identification basis

Let $e_{i_p}$ denote the standard basis for $T_p(\mathbb{R}^n)$. Theres a vector space isomorphism between $T_p(\mathbb{R}^n)$ and $D_p(\mathbb{R}^n)$, where $D_p$ is the set of derivations at $p$, ...
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### Show that S is an embedded $k$-submanifold of $M$

Assume $M$ is a smooth manifold and $S$ is a subset of $M$ such that each point $p\in S$ has a neighborhood $U \subseteq M$ such that $S\cap U$ is an embedded $k$-submanifold of $U$. I want to show ...
Let $M$ be a smooth $n-$dimensional manifold and $\alpha\in\Lambda^1(M)$ a smooth $1-$form. Let $N\subset M$ be an embedded submanifold where $\alpha|_N=0$. If $d\alpha=0$ on the whole $M$, can I say ...