# 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].

750 questions

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### Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...

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### Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

How would one solve the following orthogonal manifold problem?
$$\begin{array}{ll} \text{maximize} & \mbox{tr}(X^\top A X - X^\top B)\\ \text{subject to} & X^\top X = I\end{array}$$
where $A ...

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### Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...

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### Smooth vector fields on a surface modulo diffeomorphisms

Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...

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238 views

### Integration over a Surface without using Partition of Unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...

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206 views

### On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

I am confused with the following argument. I know I am doing something wrong but I can't find my mistake.
On one hand, one knows that if $M$ is a Lie group, then
$$\mathrm{Diff}(M)\simeq M\times\...

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46 views

### Traceless sobolev forms on compact manifolds with boundary

Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...

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328 views

### Metrics on derived smooth manifolds

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection.
For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or ...

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255 views

### History of the definition of smooth manifold with boundary

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...

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73 views

### Quartic link in a 5-sphere

In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...

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### Triple link in a 5-sphere — Proposal

In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...

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**1**answer

171 views

### Are complete minimal submanifolds closed?

Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?

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85 views

### Condition on a Lie groupoid to be represented by manifold/group or an action groupoid

Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.
When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...

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308 views

### Are framed manifolds cubulatable?

Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...

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### Smooth embedding of space forms in the Euclidean space

I was wondering which $S^n/\Gamma$ can be smoothly embedded into $\mathbb R^{n+1}$, where $\Gamma \subset O(n+1)$ is a finite subgroup. To my knowledge, the case $n \le 3$ is known. It has been proved ...